Dynamical systems and small divisors: Lectures CIME school

Perturbations of linear quasiperiodic system L. Hakan Eliasson 1-60 KAM-persistence of finite-gap solutions Sergei B...

Author: Hakan Eliasson | Sergei Kuksin | Stefano Marmi | Jean-Christophe Yoccoz | Stefano Marmi | Jean-Christophe Yoccoz

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Perturbations of linear quasiperiodic system

L. Hakan Eliasson

1-60

KAM-persistence of finite-gap solutions

Sergei B. Kuksin

61-123

Analytic linearization of circle diffeomorphisms

Jean-Christophe Yoccoz

125-173

Some open problems related to small divisors

Stefano Marmi and JeanChristophe Yoccoz

175-191

Perturbations of linear quasi-periodic system L. H. Eliasson

1

Introduction

Existence of both Floquet and l2 solutions of linear quasi-periodic skewproducts can be formulated in terms of linear operators on l2 (Z), i.e. ∞dimensional matrices. In the perturbative regime these matrices are perturbations of diagonal matrices and the problem is to diagonalize them completely or partially, i.e. to show that they have some point spectrum. The unperturbed matrices have a dense point spectrum so that their eigenvalues are, up to any order of approximation, of infinite multiplicity, which is a very delicate situation to perturb. For matrices with strong decay of the matrix elements off the diagonal this difficulty can be overcome if the eigenvectors are sufficiently well clustering. One way to handle this is to control the almost multiplicites of the eigenvalues. The eigenvalues are given by functions of one or several parameters and in order to control the almost multiplicities it is necessary that these functions are not too flat. Such a condition is delicate to verify since derivatives of eigenvalues of a matrix behave very bad under perturbations of the matrix. Derivatives of eigenvalues of matrices are therefore replaced by derivatives of resultants of matrices – an object which behaves better under perturbations. If the parameter space is one-dimensional and if the quasi-periodic frequencies satisfy some Diophantine condition, then it turns out that this control of the derivatives of eigenvalues, in terms of the resultants, is not only necessary but also sufficient for the control of the almost multiplicities. If the parameter space is higher-dimensional this control is more difficult to achieve and not yet well understood. 1.1

Examples

Discrete Schr¨ odinger equations in one dimension. In strong coupling regime this equation takes the form −ε(un+1 + un−1 ) + V (θ + nω)un = Eun ,

n ∈ Z,

(1.1)

where V is a real-valued piecewise smooth function on the d-dimensional torus Td = (R/2πZ)d and ω is a vector in Rd . The constant ε is a way to measure the size of the potential V and it is assumed to be small, i.e. a large potential. E is a real parameter.

L.H. Eliasson et al.: LNM 1784, S. Marmi and J.-C. Yoccoz (Eds.), pp. 1–60, 2002. c Springer-Verlag Berlin Heidelberg 2002

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L. H. Eliasson

When ε = 0 (1.1) have for each m ∈ Z the solution um ∈ l2 (Z),

m um n = en

if we choose E = V (θ + mω) – em n is 1 when n = m and 0 otherwise. A solution u ∈ l2 (Z) is an eigenvector of the left-hand side of (1.1) interpreted as an operator on l2 (Z). Represented in the standard basis for l2 (Z) it becomes an ∞-dimensional matrix   .. .     ..   . 0 −1            + ε V (θ + nω) −1 0 −1       ..   −1 0 .   .. . over Z. This matrix has the following properties: * dependence of parameters θ ∈ Td ; * covariance with respect to the Z-action on Td θ → θ + nω – covariance means that we obtain any row/column, and hence the whole matrix, from one row/column simply by shifting the parameter θ by the group action; * dense spectrum of the diagonal part; * domination of diagonal part when ε is small; * strong decay off the diagonal. We can also notice that the matrix is symmetric and that the diagonal part is determined by the potential V while the perturbation is independent of the potential. A natural question is if this matrix has a basis of eigenvectors in l2 (Z), in which case it can be diagonalized, or if it has any eigenvectors at all. Such eigenvectors will give solutions to (1.1) which are decaying both in forward and backward time. The existence of eigenvectors for small but positive ε is a delicate question because in the unperturbed case all eigenvalues are essentially of infinite multiplicity. Moreover there are numerous examples when there are no eigenvectors: if V is constant there are no eigenvectors since the operator un → −ε(un+1 + un−1 ) is absolutely continuous on l2 (Z); if the potential is periodic, i.e. ω ∈ 2πQd , there are no eigenvectors; if the potential is nonperiodic there are examples without eigenvectors when ω is Liouville. In weak coupling we consider the equation in the form −(un+1 + un−1 ) + εV (θ + nω)un = Eun ,

n ∈ Z.

(1.2)

Perturbations of linear quasi-periodic system

3

When ε = 0 (1.2) has, for each ξ ∈ T, extended solutions uξ ∈ l∞ (Z),

uξn = einξ ,

for E = −2 cos(ξ). When ε is small it is natural to look for extended solutions of the form uξn = einξ U (θ + nω), where U : Td → C. Such solutions are known as Floquet solutions or Bloch waves. The equation for U then becomes −(eiξ U (θ + ω) + e−iξ U (θ − ω)) + εV (θ)U (θ) = EU (θ) which, when written in Fourier coefficients, gives the matrix  .. .    ..  0 Vˆ (k − l)   .       − ε 2 cos(ξ+ < k, ω >)     ..  Vˆ (l − k) 0 . 



..

         .

over Zd . This matrix has the properties: * dependence of parameters ξ ∈ T; * covariance with respect to the Zd -action on T ξ → ξ+ < k, ω > . It has like (1.1) a dense spectrum of the diagonal part, domination of diagonal part when ε is small and decay off the diagonal (depending on the smoothness of V ). In this case however, the matrix is complex and Hermitian, the diagonal part is fixed = 2 cos(α) and the perturbation depends on the potential function V . Finding Floquet solutions for (1.2) now amounts to finding eigenvectors of this matrix. This is a delicate matter which is known to depends on arithmetical properties of ω and requires strong smoothness of V which is reflected in strong decay off the diagonal of the matrix. Floquet solutions are related to the absolutely continuous spectrum and there are examples of discontinuous potentials with only singular spectrum. In strong coupling we can consider generalizations of the form −ε(∆θ u)n + V (θ + nω)un = Eun ,

n ∈ Z,

where the difference operator is allowed to take a more general form (∆θ u)n =

N j=−N

bj (θ + (n + j)ω)un+j .

(1.1 )

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L. H. Eliasson

It could even be infinite with sufficiently strong decay. This equation will now give rise to a perturbation problem with the same diagonal part as (1.1) but with a perturbation that depends on the bj ’s. It is typically not symmetric and it is therefore not reasonable to try to diagonalize it for small ε. But block diagonalization and construction of eigenvectors still make sense. In weak coupling we can consider equation −(∆u)n + εV (θ + nω)un = Eun ,

n ∈ Z,

(1.2 )

with a constant difference operator of the more general form N

(∆u)n =

bj un+j .

j=−N

It can also be infinite with sufficiently strong decay. It will now give rise to a perturbation problem of the same type as (1.2) but with diagonal part determined by the function N bj eijξ . j=−N

Discrete Linear Skew-Products. We consider a weakly perturbed skewproduct of the form Xn+1 − (A + εB(θ + nω))Xn = EXn ,

n ∈ Z,

(1.3)

where A ∈ Gl(N, R), B : Td → gl(N, R). When ε = 0 we have the Floquet solutions Xn = einξ X0 ,

AX0 = aX0 ,

for E = eiξ − a, ξ ∈ T. For ε = 0 we look for Floquet solutions Xn = einξ Y (θ + nω) with Y : Td → CN . The equation for Y then becomes eiξ Y (θ + ω) − (A + εB(θ))Y (θ) = EY (θ) which, when expressed in Fourier coefficients, becomes the matrix  ..   . ..  .    ˆ − l) 0 B(k      − ε ei(ξ+) I − A ˆ − k)    B(l 0  .. . over Zd . The properties of the matrix are:



..

      .


5

* dependence of parameters ξ ∈ T; * covariance with respect to the Zd -action on T ξ → ξ+ < k, ω > . The matrix has dense point spectrum of diagonal part, domination of diagonal part when ε is small and decay off the diagonal (depending on the smoothness of B). Notice that here the unperturbed part is only block diagonal and its spectrum is determined by the functions (eξ − aj ) where aj runs over the eigenvalues of A – a multi-level operator. Finding Floquet solutions for (1.3) now amounts to finding eigenvectors of this matrix. We shall give two example of skew-products which can be considered as strongly perturbed. The first example is an obvious generalization of the Schr¨ odinger equation: −ε(Xn+1 + Xn−1 ) + B(θ + nω)Xn = EXn ,

n ∈ Z,

(1.4)

where B : Td → gl(N, R) is a symmetric matrix. This now gives rise to a multi-level version of (1.1). The second example arises from the continuous Schrödinger equation in strong coupling −εu (t) + V (θ + tω)u(t) = Eu(t),

t ∈ R.

(1.5)

For ε = 0 the operator to the left becomes multiplication by V (θ + tω) which has purely continuous spectrum and no eigenvalues. Hence this is an even more complicated object to perturb from. Instead we consider the corresponding dynamical system u v

=

0 1 ε (V

u 1

(θ + tω) − E) 0

v

,

θ = θ + ω,

and its time-t-map (Φt (θ; E, ε), θ + ω). Let ω = (ω , 1) and θ = (θ , θd ). Then the evolution of the system is described by un+1 an bn un un = =: Φ1 ((θ + nω , 0); E, ε) , vn+1 cn dn vn vn where an dn − bn cn = 1. From this we deduce the second order equation bn−1 un+1 + bn un−1 − W (θ + nω )un = 0, where W (θ ) = a0 (θ )b0 (θ − ω ) + d(θ − ω )b(θ ).

(1.6)

6

L. H. Eliasson

In matrix formulation the right-hand side of (1.6) becomes  .. .    ..  0 bn−2    .      +   bn 0 bn−1 W (θ + nω )    ..  bn+1 0 . 



..

    .     .

which is a matrix over Z with properties: * dependence of parameters θ ∈ Td−1 ; * covariance with respect to the Z-action on Td−1 θ → θ + nω . The diagonal part has dense spectrum and the matrix has strong decay off the diagonal. There is some domination of the diagonal part if ε is small and if E is close to inf V but it is less explicit than in (1.1). Both the diagonal and the non-diagonal part depend on the potential V . Since the matrix is neither symmetric nor Hermitian we cannot hope to diagonalize it, but it may still be block diagonalizable and have a pure point spectrum with finite multiplicity. If 0 is an eigenvalue for a given parameter value E, then the corresponding eigenvector will give a solution of (1.5) which is decaying both in forward and backward time giving some point spectrum for that operator. Higher-dimensional Schr¨ odinger equations. We can consider the Schrödinger equation in, say, two dimensions in strong coupling −ε(∆u)n + V (θ1 + n1 ω1 , θ2 + n2 ω2 )un = Eun ,

n ∈ Z2 ,

(1.7)

where ∆ is the nearest neighbor difference operator (∆u)n = um , |m−n|=1

V is a real valued piecewise smooth function on the product of two tori Td ×Td and ω = (ω1 , ω2 ) is a vector in Rd × Rd . A solution u ∈ l2 (Z2 ) is an eigenvector of a perturbation of the matrix   .. .       V (θ1 + n1 ω1 , θ2 + n2 ω2 )   .. . over Z2 . This matrix has the following properties:


7

* dependence of parameters θ ∈ T2d ; * covariance with respect to the Z2 -action on T2d θ = (θ1 , θ2 ) → (θ1 + n1 ω1 , θ2 + n2 ω2 ). The diagonal part has dense spectrum and dominates the matrix when ε is small, and the decay off the diagonal is strong. In weak coupling −(∆u)n + εV (θ1 + n1 ω1 , θ2 + n2 ω2 )un = Eun ,

n ∈ Z2 ,

(1.8)

the existence of Floquet solutions un = ei(n1 ξ1 +n2 ξ2 ) U (θ1 + n1 ω1 , θ2 + n2 ω2 ) gives rise to an eigenvalue problem in the same way as (1.2). 1.2

Two basic difficulties

The fundamental problem in the perturbation theory of these matrices is that all eigenvalues have infinite multiplicity up to any order of approximation. More precisely, we say that two eigenvalues E m (θ) and E n (θ) are ρ-almost multiple if | E m (θ) − E n (θ) | ≤ ρ. Then the ρ-almost multiplicity is infinite for any ρ = 0. This is not only a technical obstacle since for example the matrix derived from equation (1.1) does not have any eigenvectors for any ε = 0 if V is constant. Infinite almost-multiplicity can be handled if the matrix has strong decay off the diagonal and if the corresponding almost-multiple eigenvectors {umi } cluster. By this we mean that the range of all the (umi )’s, i.e. the infinite subset i ∪i {n ∈ Z : um n = 0} collects into finite clusters in Z. This clustering of almost-multiple eigenvectors is crucial for the possible diagonalization of the perturbed matrix. It should be measured by the separation and extension of the clusters – which are related to the decay of the matrix off the diagonal – and by the the cardinality of the clusters. The first difficulty is therefore to get good clustering of eigenvectors. An essential role in this is played by the strong decay of the perturbation off the diagonal and by the Diophantine properties of the frequency vector ω. A vector ω ∈ Rd is said to be Diophantine with parameters κ, τ > 0 if inf |< n, ω > −2πm | ≥

m∈Z

κ , | n |τ

∀n ∈ Zd \ 0.

(1.9)

8

L. H. Eliasson

Control of the almost multipicities and clustering. In the problem (1.1) the eigenvalues E m (θ) and the eigenvectors um (θ) are E m (θ) = V (θ + mω)

m and um n = en .

Which are the eigenvalues that are ρ-almost multiple to E m (θ) and how many are there in any finite subset of Z? In order to investigate this question, consider the set ∆ = {x ∈ T :| E m (θ + x) − E m (θ) |≤ ρ}. If V is smooth, measured by β and γ, | V |C k ≤ β(k!)2 γ k

∀k ≥ 0,

and if V satisfies the transversality condition, measured by σ and s, max | ∂xk (V (θ + x) − V (θ)) | ≥ σ

0≤k≤s

∀θ, x,

then ∆ (see Lemma B1) is a union of intervals ∪∆j with a finite number ≤ µ (independent of ε) of components, and each component is contained in an interval with length 1 ≤ const ρ s – when V (θ) = cos(θ) the numbers µ and s are both 2. If there are more than µ + 1 eigenvalues E nj (θ), j = 1, 2, . . . , µ + 1, that are ρ-almost multiple to E m (θ), then it follows that θ + ni ω and θ + nj ω belong to one component, ∆1 say, for at least two different i, j ≤ µ + 1. What controls the return of a translation θ → θ + ω to an interval is the Diophantine condition (1.9). Under this condition it follows that 1 1 1 | ni − nj |≥ constκ τ ( ) τ s , ρ

which means that when ρ is small then the distance between the sites ni and nj is very large. Consider now a sequence of eigenvalues E m (θ) = E n1 (θ), E n2 (θ), . . . , nµ+1 E (θ) such that any two consecutive pairs are ρ-close. Then any two pairs are (µ + 1)ρ-close and hence it must hold that max

1≤i,j≤µ+1

1

| ni − nj |≥ ν = constκ τ (

1 1 ) τs . (µ + 1)ρ

This implies that the ranges of the corresponding eigenvectors uni , i.e. the set of indices or sites {ni }, collects into clusters in Z containing at most µ many sites. Any two clusters are separated by at least ν many sites, and each cluster extends over at most λ = µν sites.


9

This simple argument which controls the clustering of the eigenvectors does not depend on the dimension of the lattice Z, but it does depend crucially on the dimension of the parameter space {θ}. If we consider V (θ) = cos(θ1 ) + cos(θ2 ), defined on the 2-dimensional torus, with ω = (ω1 , ω2 ), then the corresponding set ∆ is now a subset of T2 and its components ∆i are simply connected subsets of T2 whose shape depend on the potential V . A Diophantine condition on ω does not alone control the return of a translation to an open connected set – the return also depend on the geometry of the sets ∆i . Of course the situation is the same if we consider a truly higher-dimensional problem like (1.7) or (1.8). This is why the higher-dimensional case is more difficult and why our results in this paper are restricted to matrices which are covariant with respect to a group action on the one-dimensional space T1 . Propagation of smoothness and transversality. The second difficulty is that the diagonalization is a small divisor problem which requires a quadratic iteration of KAM-type. It is therefore not enough to control the almostmultiplicities of the unperturbed matrix but they must be controlled also for nearby matrices – if not for all so at least for a sufficiently large class of nearby matrices. This class of matrices is the normal form matrices: their elements have exponential decay off the diagonal measured by a parameter α; their smoothness is measured by β, γ; they are not smooth on the whole parameter space but only piecewise smooth over a partition P. They will have a pure point spectrum and will be characterized by a certain block structure – measured by λ, µ, ν, ρ – which describes their eigenvectors. However, they will not be diagonal nor block diagonal in general. Since derivatives of eigenvalues behave very bad under perturbations we cannot hope that the transversality condition of the eigenvalues of the unperturbed matrix propagates to the eigenvalues of the nearby normal form matrices. But instead of comparing two eigenvalues we can compare the whole spectrum of two submatrices truncated over blocks. This is done by the resultant and what we need is a transversality condition of the resultants of submatrices truncated over blocks – transversality of blocks. In distinction to eigenvalues, derivatives of resultants behave nicely under perturbations and we are able to show that this transversality condition of a normal form matrix propagates to nearby normal form matrices if their block structures are compatible in a certain sense. 1.3

Results and References

Some attempts to prove pure point spectrum for the discrete quasi-periodic Schr¨ odinger equation (1.1) on a one-dimensional torus in strong coupling were made already in the early 80’s [1,2] but the breakthrough came in the late 80’s with the work of Sinai [3] and of Fr¨ ohlich, Spencer and Wittver [4]

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L. H. Eliasson

on a “cosine”-like potential. The most general result today is [5]. The present article provides a different presentation of the result in [5], together with generalizations and variants. Similar results have also been obtained in [6] by different methods. Discrete equations (1.1 ) were treated in [5,7]. In this paper we generalize the these results still a bit more. It should be noted the we only discuss the perturbation theory for smooth systems with Diophantine frequencies. There are numerous results for systems that are non-smooth or have non-Diophantine frequencies. For example, the results referred to above about Liouville frequencies and discontinuous potentials can be found in [8,9]. The results on weak coupling (1.2) are older. Dinaburg and Sinai [10] constructed Floquet solutions for the continuous Schr¨ odinger in order to get some absolutely continuous spectrum (see also [11]). These results were then extended in [12] and the proof that there is such a solution for almost all quasi-momentum was given in [13]. The work [13] even proves that the oneparameter family (1.2) have Floquet solutions for a.e. value of the spectral parameter E. All these results were proved for the continuous equation but the adaptation to the discrete case is likely to be quite straight forward. The results of [10] was for example carried over to the discrete case in [14]. The results [10–14] were proven by ODE-methods. An operator approach like the one in this paper were used in [15] and later also in [16]. For more general weakly perturbed skew-products (1.3) results have been obtained in [17–19], also with the use ODE-methods. (See also the short survey [20].) Some further results are obtained in this paper. The continuous Schr¨ odinger in strong coupling (1.5) has only been treated in one particular case [4]: −εu (t) + (cos(θ1 + t) + cos(θ2 + tω))u(t) = Eu(t). This equation has, for ε small enough, a pure point spectrum close to the bottom of the spectrum. (From [13] it follows that it has purely absolutely continuous spectrum with Floquet solutions for sufficiently large E.) Other strongly perturbed skew-products have been studied in [21] from the point of view of Lyapunov exponents. For equation (1.1) on a higher-dimensional torus there are two papers [22,23] which, in particular, stresses the importance of “external” parameters in these problems. In truly higher dimension essentially nothing is proven. For the continuous analogue of (1.8) some results, also with the use of “external” parameters, have been announced [24,25]. The difficulties met in higher dimensions are related to those in other problems on infinite-dimensional matrices [26]. In this paper we only discuss discrete quasi-periodic skew-products. The continuous case is both similar and different. The construction of Floquet solution does give rise to a similar perturbation problem. The difference is that the parameter space is the Euclidean space R, instead of the compact


11

circle T, and that the spectrum of the diagonal part is unbounded. This is likely to be a minor point and we believe that all the results in weak coupling carry over from discrete to continuous skew-products. The construction of l2 solutions of continuous skew-products is, however, much more difficult since it does not give rise to such a nice perturbation problem as in the discrete case. 1.4

Description of the paper

In Section 1 we introduce some notations. In Section 2 we define the normal form matrices which are a sort of generalized block matrices. They depend smoothly on some parameters and are covariant with respect to a group action. The smoothness is only piecewise in a way we explain. In Section 3 we show that we can conjugate the normal form matrices to block diagonal form, but there is a prize to pay – the smoothness properties of the conjugated matrix is less good. In Section 4 we show that we can conjugate a perturbation of a normal form matrix to a new normal form matrix with a much smaller perturbation. This should be the starting point for an iteration of KAMtype. In Section 5 we discuss under what conditions this procedure can be iterated and the role of clustering of the blocks. Up to this point neither the smoothness nor the group action play any role whatsoever. The clustering property is related to almost multiplicities of the eigenvalues. These multiplicities are related to estimates of resultants which can be obtained from a transversality property. In Section 6 we discuss when and how this transversality property of the resultants can be transferred to nearby normal form matrices. In Section 7 we specialize to a quasi-periodic group action on the onedimensional torus, i.e. to a linear ergodic action on T, submitted to a Diophantine condition. In this case the transversality property of the resultants will be sufficient to control the almost multiplicities of the eigenvalues and hence the clustering of the blocks. This will give us a perturbation theorem, Theorem 12, of a quite general type. In Section 8 we discuss applications of Theorem 12 to some of the one-dimensional problems we have discussed above. In an appendix we include basic finite-dimensional results from analysis and linear algebra. There is some difference, besides presentation, from the approach in [5]. We work with Gevrey functions – the particular Gevrey class has no importance – but instead of using the possibility to smoothly truncate Gevrey function as in [5] we use here discontinuous cut-offs. The disadvantage is that we have to work with piecewise smooth functions, which is a little awkward in relation to the covariance property, and that we must control the size of the pieces. The advantage is that the particular group action does not intervene before the final step of the proof.

12

1.5

L. H. Eliasson

Notations

Let L be a standard lattice Z, Z2 , Z3 , . . . and denote its elements, called indices or sites, by a, b, c, . . . . If Ω, Ω ⊂ L, then we let dist(Ω, Ω ) =

min

a∈Ω,b∈Ω

| a − b |,

where | | denotes the Euclidean distance. We let {eb : b ∈ L} denote the standard basis of l2 (L), defined by eba = 1 or 0 depending on if a = b or not. If Ω is a subset of L, we denote by CΩ the subspace of l2 (L) spanned by {ea : a ∈ Ω}. A subspace Λ of dimension k of l2 (L) comes equipped with an ON-frame, i.e. Λ is a linear mapping Λ : Ck → l2 (L) such that

Λ∗ Λ = IdCk ΛΛ∗ = ⊥ −projection of l2 (L) onto Λ.

The range of Λ is the set of sites that Λ occupies, i.e. the set of all a ∈ L such that (ea )∗ Λ = 0. A block for Λ is a subset Ω ⊂ L that contains the range of Λ. If D : l2 (L) → l2 (L) is a bounded operator we identify it with an ∞dimensional matrix via the standard basis: Dab = (ea )∗ Deb , the matrix element at row a and column b. By a matrix on L we shall always understand such a representation of a bounded operator. The truncation at ˜ defined by D ˜ b = Db or = 0 distance N off the diagonal of D is the matrix D a a depending on if | b − a |≤ N or not. If Λ is a subspace we let DΛ = Λ∗ DΛ and if Ω ⊂ L we denote by DΩ the matrix DCΩ . A partition of a manifold X is a (locally finite) collection of open subsets – pieces – P such that ∪Y ∈P Y is of full measure in X. If P and Q are two such decomposition and if T is a homeomorphism on X, then

P ∨ Q = {Y ∩ Z : Y ∈ P, Z ∈ Q} T (P) = {T (Y ) : Y ∈ P}. Td denotes the d-dimensional torus (R/2πZ)d and its distance is denoted by . We denote by | u | the Euclidean norm if u is a vector and the operator norm if u is a linear operator. For a smooth function u (possibly vector or matrix valued) defined in an open box V in Rd we let | u |C k be sup sup | 0≤l≤k x∈V

1 l ∂ uj (x) |, (l!)2

k ∈ Nd .


13

u is said to be of Gevrey class G2 if | u |C k ≤ βγ k

∀k ≥ 0.

Smooth function will always be assumed to be G2 unless otherwise specified – that we have chosen this particular Gevrey class will be of no importance. A function is said to be smooth on P if it is smooth on (a neighborhood of the closure of) each piece of P.

2 2.1

Covariance and normal form matrices Covariance

Let L be a lattice Z, Z2 , . . . and let X be a a torus T, T2 , . . . . On the torus X we have an L-action T : L × X → X, and we denote by Ta the mapping x → T (a, x), a ∈ L, x ∈ X. For an element b of L, define τb : l2 (L) → l2 (L) by (τb f )a = fa−b . Clearly τb is unitary and τb∗ = τb−1 = τ−b . Definition. A matrix D : l2 (L) × X → l2 (L) is covariant with respect to the group action T if τc∗ D(x)τc = D(Tc (x))

∀ c ∈ L, x ∈ X.

A covariant matrix D(x) is pure point if, for almost all x, there exists an eigenvector q(x) such that {q a (x) =: τa q(Ta (x)) : a ∈ L} is a basis for l2 (L). Ω(x) ⊂ L is a block for q(x) if Ω(x) ⊃ {a ∈ L : (ea )∗ q(x) = 0}. If q(x) is an eigenvector with eigenvalue E(x) and block Ω(x), q b (x) = τb q(Tb (x)) is an eigenvector with eigenvalue E b (x) = E(Tb (x)) and block Ωb (x) = Ω(Tb (x)) − b. In terms of the matrix elements, covariance means that b+c (x) = Dab (Tc x) Da+c

∀a, b, c ∈ L, x ∈ X.

More generally, we say that D(x) is pure point if q(x) is a generalized eigenvector with finite multiplicity. A block for q(x) is then a block for the smallest invariant space containing q(x).

14

L. H. Eliasson

Even more generally we can consider multi-level operators D : l2 (L) ⊗ CN × X → l2 (L) ⊗ CN . On l2 (L) ⊗ CN we have the standard basis {eb,j : b ∈ L, j = 1, . . . , N } defined by eb,j a,i = 1 or 0 depending on if (a, i) = (b, j) or not. For an element b ∈ L, we define τb : l2 (L) ⊗ CN → l2 (L) ⊗ CN by (τb f )a,i = fa−b,i ,

i = 1, . . . , N.

The covariance of D with respect to the L-action T is now defined in the same way as above. D(x) is pure point if l2 (L) ⊗ CN has a basis {q b,j (x) : a ∈ L, j = 1, . . . N } of eigenvectors of D(x). A block for q j is a subset Ω(x) ⊂ L × {1, . . . , N } such that Ω(x) ⊃ {(a, i) ∈ L × {1, . . . , N } : (ea,i )∗ q j (x) = 0}. We can also consider in the multi-level case generalized eigenvectors of finite multiplicity q j (x). 2.2

Normal Form Matrices

We shall consider matrices D : l2 (L) × X → l2 (L) which are covariant with respect to an L-action T :L×X →X and which satisfy the following conditions. Exponential decay off the diagonal. | Dab |C 0 ≤ βe−α|b−a| .

(2.1)

Smoothness. The components of D are piecewise smooth and satisfy | Dab |C k ≤ βe−α|b−a| γ k

∀k ≥ 1.

(2.2)

D(x) is pure point with a (possibly generalized) eigenvector q(x), corresponding eigenvalue E(x) and block Ω(x) satisfying the following conditions. Block dimensions and block extensions. For all x ∈ X

Ω(x) ⊂ {a :| a |≤ λ} (2.3) #Ω(x) ≤ µ.


Block overlapping. For all x ∈ X #

Ωa (x) ≤ µ.

15

(2.4)

Ωa (x)∩Ω(x)=0

If the blocks do not overlap, then the matrix D is a block matrix with blocks of dimension ≤ µ. In general the blocks do overlap but we shall require that resonant blocks don’t. Resonant block separation. For all x ∈ X | E a (x) − E(x) |≤ ρ =⇒ Ωa (x) = Ω(x) or dist(Ωa (x), Ω(x)) ≥ ν ≥ 1

(2.5)

Partition. There is a locally finite partition P such that D and Ω are smooth on the partition P.

(2.6)

By a matrix D being smooth on a partition we mean that each Dab is smooth on, a neighborhood of the closure of, the pieces of ∨{Tc (P) :| c +

b−a b+a |≤| |}. 2 2

By a set Ω being smooth on a partition we mean that each the characteristic function χΩb (a) is smooth on, a neighborhood of the closure of, the pieces of the same partition. This smoothness condition is consistent with the covariance. Remark. This concept of piecewise smoothness in relation to covariance is a little awkward. If A and B are two covariant matrices which are smooth on the partition, then AB will not be smooth on any partition unless A or B are truncated at some finite distance from the diagonal. If A or B are truncated at distance N , then AB is smooth on the refined partition P(N ) =: ∨{Tc (P) :| c |≤ N }. Matrices like eA or (I + A)−1 will in general not be be piecewise smooth with the definition we have chosen. Given a partition P we let Pδ denote another type of refinement. Each piece Y of Pδ is contained in a piece of P and has “diameter” less than δ, i.e. any two points x, y in Y can be joined by a curve in Y of length less than δ – if Y is convex then δ is the usual diameter of Y . In the sequel we shall construct partitions like P(N1 )δ1 (N2 )δ2 (N3 ) . . . .

16

L. H. Eliasson

Definition. We say that a covariant matrix D is on normal form N F(α, β, γ, λ, µ, ν, ρ) if both D and D∗ satisfy (2.1 − 6) with the same block Ω and partition P. ¯D . The eigenvectors qD The eigenvalues will be complex conjugate, ED∗ = E and qD∗ of course not related unless D is symmetric, in which case they are equal, or Hermitian, in which case the are complex conjugate. We also use notations like N F(α, β, γ, λ, µ, ν, ρ; q, E, Ω, P) when we want to stress these objects. Remarks. 1. The parameters β, γ, λ, µ can be increased and the parameters α, ν, ρ can be decreased. For simplicity we shall assume that β, γ both are ≥ 1. 2. It follows from the definition that D is truncated at distance 2λ from the diagonal and that, for given a (or b), Dab = 0 for at most µ2 many b’s (or a’s). 3. Using the generalized Young inequality [27] we get an estimate of D in the operator norm | D |C k ≤ β(

eα + 1 dim L k ) γ eα − 1

∀k ≥ 0.

(2.7)

If D is a m × m-dimensional matrix then we also have | D |C k ≤ mβγ k

∀k ≥ 0.

(2.8)

The estimate (2.8) is better than (2.7) if α is small but worse if m is large. We have the corresponding notion for multi-level operators on l2 (L) ⊗ CN . We then have N eigenvectors q j , each with corresponding eigenvalue E j and b,j block Ωj . (2.1-2) should hold for the matrix elements Da,i and (2.3) should hold for the blocks. (2.4) and (2.5) takes the form: for all j and for all x ∈ X

#

Ωa,i (x) ≤ µ;

(2.4 )

Ωa,i (x)∩Ωj (x)=0

for all i, j and for all x ∈ X | E a,i (x) − E j (x) |≤ ρ =⇒ Ωa,i (x) = Ωj (x) or dist(Ωa,i (x), Ωj (x)) ≥ ν ≥ 1.

(2.5 )


3

17

Block splitting

In this section we shall see that we can conjugate a normal form matrix to a true block diagonal matrix. The price to pay is in the smoothness and will be related to almost-multiplicity of the eigenvalues. Proposition 1. Let D ∈ N F(α, β, γ, λ, µ, ν, ρ; Ω, P). Then there exists a subspace Λ(x) which is invariant under D(x) such that all eigenvalues of Λ∗ DΛ are ρ-close, and which has the following properties: (i)

for all x Λ(x) ⊂ CΩ(x) ;

(3.1)

(ii) | Λ |C k ≤ ((const

βµ2 2µ k ) γ) ρ

∀k ≥ 0;

(3.2)

(iii) Λ is smooth on the partition P(3λ)δ , where δ = (const

ρ 2µ 1 ; ) βµ2 γ

(3.3)

(iv) the angle between Λ(x) and the invariant subspace Λb (x), Λb (x) = τb Λ(Tb x), Λb (x)=Λ(x)

is ≥ (const

ρ µ2 ) . βµ2

(3.4)

The constants only depend on the dimensions of L and X. ˜ Proof. Let Ω(x) = ∪Ωa (x) where the union is taken over all a such that a Ω (x) ∩ Ω(x) = ∅. Then by (2.3 − 4), for each x,

˜ #Ω(x) ≤µ ˜ Ω(x) ⊂ {a ∈ L :| a |≤ 3λ}. Consider DΩ˜ . This matrix is smooth on P(3λ) and by the remark (2.8) we have the estimate | DΩ˜ |C k ≤ µβγ k ∀k ≥ 0.

18

L. H. Eliasson

Apply Lemma A.6 with ρ = r to get a DΩ˜ (x)-invariant decomposition ˜

CΩ(x) =

k

˜ i (x) Λ

i=1

smooth on P(3λ)δ , with δ satisfying (3.3), and 2

˜ i |C k ≤ ((const βµ )2µ γ)k |Λ ρ

∀k ≥ 0.

˜ 1 (x) say. Let now x be fixed. If q˜ is q(x) belongs to one of these spaces – Λ ˜ 1 (x) then it follows from Lemma A.7 an eigenvector of DΩ˜ (x) which lies in Λ that either q˜ is an eigenvector of D(x) or is perpendicular to CΩ(x) . Hence, if qã = 0 for some a ∈ Ω(x), then q˜ is an eigenvector of D(x) and hence = q b (x) for some b. This implies that | E(x) − E b (x) |≤ ρ and by (2.5) that Ωb (x) = Ω(x). Hence, either q˜ is supported in Ω(x), in which case it is an eigenvector of D(x), or it is supported in the complement of Ω(x). Therefore ˜ 1 (x) is an invariant space for D(x) which satisfies the same Λ(x) = CΩ(x) ∩ Λ ˜ 1 , i.e. (3.2-3). This proves (i), (ii) and (iii). estimates as Λ The angle between Λ and Λb follows also from Lemma A.6 since each ˜ Λb (x), which is not orthogonal to Λ(x), is contained in C Ω(x) and therefore ˜ i. is a subspace of some Λ When D is Hermitian in Proposition 1, then the angle between the invariant spaces is π2 ; the radius δ in (3.2) is const

ρ . βγµ2

In the multi-level case we get one subspace Λj (x) for each j = 1, . . . , N with j Λj (x) ⊂ CΩ (x) , and the matrix Q is defined by Q(x) = (. . . Λb (x) . . . ), where Λb is the subspace (Λb,1 , . . . , Λb,N ). All the estimates are the same and the proof is also the same with obvious modifications.

4

Quadratic convergence

In this section we shall construct a conjugation which transforms a normal form plus a perturbation to a new normal form plus a smaller perturbation. The formulation will involve an auxiliary matrix W . The reason for this matrix is that we are not allowed to take inverses because they are not piecewise smooth, but we have to use approximate inverses. The role of W is to measure this approximation.


19

Lemma 2. Let D ∈ N F(α, . . . , ρ; E, P) and let F be a covariant matrix, smooth on P and satisfying | Fab |C k ≤ εe−α|b−a| γ k

∀k ≥ 0.

(4.1)

Then there exist covariant matrices K and G satisfying

[D, K] = F − G a b < qD if | E a (x) − E b (x) |> ρ, ∗ (x), G(x)qD (x) >= 0 with the following properties: (i)

for all k ≥ 0

2 3µ3 6λα −α|b−a| k | Kab |C k + | Gba |C k ≤ ε( βµ1 2 )(const βµ e e (γ ) ρ ) 2

3µ γ = (const βµ ρ )

3

+4µ2

γ

(4.2)

(ii) K and G are smooth on the partition P(3λ)δ (6λ) where δ = (const

ρ 2µ 1 ; ) βµ2 γ

(4.3)

(iii) if F is truncated at distance λ from the diagonal, then K and G are truncated at distance λ + 6λ. The constants only depend on the dimensions of L and X. Proof. Let Λ be the invariant space of D defined in Proposition 1, with ρ replaced by ρ3 , and let Q = (. . . Λa Λb . . . ). The estimate of Q follows from that of Λ, and Q is truncated at distance λ from the diagonal. Let ˜b . . . ) ˜ = (. . . Λ ˜ aΛ Q ˜∗, be the same matrix as Q but obtained from D∗ . The inverse Q−1 = B Q where B is a block matrix with the blocks ˜ a )∗ Λa ]−1 . [(Λ It follows that Q−1 is truncated at distance 2λ from the diagonal, is smooth on P(3λ)δ (λ) and satisfies | Q−1 |C k ≤ (const – see Lemma A.6.

βµ2 2µ2 βµ2 4µ2 k ) ((const ) γ) ρ ρ

∀k ≥ 0

20

L. H. Eliasson

˜ = Q−1 DQ is a block diagonal matrix with blocks – which we Then D ˜ a – corresponding to the subspaces Λa , which is smooth on here denote by D a P(3λ)δ (3λ) and satisfies 2

2

˜ b |C k ≤ βµ(const βµ )2µ2 e3λα e−α|b−a| ((const βµ )4µ2 γ)k |D a ρ ρ

∀k ≥ 0.

˜ a and D ˜ b have two eigenvalues that ˜ a are ρ -close, so if D Eigenvalues of D a a b 3 ˜ a and D ˜ b respectively differ differ by at least ρ then any two eigenvalues of D a b ρ by at least 3 . ˜ Let now F˜ = Q−1 F Q. Then F˜ will be smooth on the same partition as D and satisfy the same estimate but with the first factor β replaced by ε. The equation becomes ˜b ˜b ˜b ˜b ˜ aK ˜b D a a − Ka Db = Fa − Ga . ˜ b respectively differ by ≤ ρ, then the ˜ a and D If any two eigenvalues of D a b equation reduces to ˜ b = 0 and G ˜ b = F˜ b , K a a a and if at least two eigenvalues differ by > ρ, then the equation becomes ˜ bD ˜b ˜ ab − K ˜b ˜ aa K D a b = Fa

˜ b = 0. and G a

˜ is clear so we only need to estimate K ˜ from the second The estimate of G ˜ a by a unitary transformation we equation. Since we can triangularize each D a get 2 ˜ b |C 0 ≤ ε[ βµ (const βµ )2µ2 ]µ e3λα e−α|b−a| . |K a ρ ρ Since the solution of the second equation is unique we get estimate of the derivatives by differentiating the equation. ˜ −1 and G = QGQ ˜ −1 . They will satisfy the required We now let K = QKQ estimates and be smooth on P(3λ)δ (6λ). (iii) is clear by construction. Proposition 3. Let D ∈ N F(α, . . . , ρ; Ω, P) and let F and W be covariant matrices, smooth on P and satisfying | Fab |C k + | Wab |C k ≤ εe−α|b−a| γ k

∀k ≥ 0.

Then there exists a constant C – depending only on the dimensions of X and L – such that if ε ≤ ξ(

ρ − ρ (α − α )dim L )µ , β

then there exist

ξ =: (C

ρ 3µ3 −8λα ) e , βµ2

D ∈ N F(α , β , γ , λ, µ, ν, ρ ; Ω, P )

(4.4)


21

and covariant matrices U , V , F and W such that V (D + F )U = D + F

and

V (I + W )U = I + W

with the following properties: (i)

α < α, 2

3µ γ = ( C1 βµ ρ )

3

+4µ2

β = (1 + ξε )β,

γ,

ρ < ρ;

(4.5)

(ii) for all k ≥ 0, | (U − I)ba |C k + | (V − I)ba |C k + | (D − D)ba |C k ≤

ε −α|b−a| k e (γ ) , ξ (4.6)

and D − D is truncated at distance ν from the diagonal; (iii) for all k ≥ 0,

| (F )ba |C k + | (W )ba |C k ≤ ε e−α |b−a| (γ )k 2 ε −(ν−8λ)(α−α ) ε = max[ ξ2 (α−α ]; )dim L , εe

(4.7)

(iv) F and W are smooth on P = P(5λ)δ (6λ + 2ν) where δ = (C

ρ 2µ 1 . ) βµ2 γ

(4.8)

Notice that the smallness assumption only depends on α, β, λ, µ, ρ, α − α , ρ − ρ and that the size of the new perturbation F depends on ν. Nothing depends on γ. ˆ be the truncations of F and W at distance ν − 8λ from Proof. Let Fˆ and W the diagonal. Define K and G so that 1 ˆ ˆ ). D + DW [K, D] + G = Fˆ − (W 2 (This choice gives an Hermitian G and an anti-Hermitian K whenever D, F, W are Hermitian.) The right hand side is smooth on P(2λ), truncated at distance ν − 6λ from the diagonal and satisfies | (RHS)ba |C k ≤ constεβµ2 e2λα e−α|b−a| γ k

k ≥ 0,

because D is truncated at distance 2λ and Dcd = 0 for at most µ2 many c’s or d’s.

22

L. H. Eliasson

Hence, by Lemma 2 we have | Kab |C k + | Gba |C k ≤

ε −α|b−a| k e (γ ) ξ

and K and G are smooth on the partition P(5λ)δ (6λ). Moreover, K and G are both truncated at distance ν. If we define 1 ˆ , U =I +K − W 2

1 ˆ V =I −K − W 2

then U, V and G will satisfy the estimate (4.6) and ˆ ) + O2 (W, W ˆ , K)] + [O3 (W, W ˆ , K)] V (I + W )U − I = [(W − W =: W = W2 + W3 and ˆ , K)] + [O3 (F, W ˆ , K)] V (D + F )U − D − G = [(F − Fˆ ) + O2 (F, W =: F = F2 + F3 . Estimating this using Lemma A.8 gives that Fj and Wj satisfies (4.7). Observing that W and F are at most linear in the non-truncated matrices W, F we see that they are smooth on the partition P(5λ)δ (6λ + 2ν). It is clear that D = D + G satisfies (2.1-4) with α , β , γ , λ, µ and (2.6) with P Since the eigenvalues of DΩ and DΩ differ by at most 4(const

β αdim L

1

)1− µ (const

ε ξαdim L

1

)µ ≤

1 (ρ − ρ ) 4

(Lemma A.2), where we used (2.7) to estimate the norm of the matrices, it follows that | (E )a (x) − (E )(x) |≤ ρ →| E a (x) − E(x) |≤ ρ. Therefore D verifies (2.5).

We now iterate the construction in Proposition 3 a finite number of times in order to improve the result. Corollary 4. Let D ∈ N F(α, . . . , ρ; Ω, P) and let F and W be covariant matrices, smooth on P and satisfying | Fab |C k + | Wab |C k ≤ εe−α|b−a| γ k

∀k ≥ 0.


23

Then there exists a constant C – depending only on the dimensions of X and L – such that if

dim L µ ) ] ε ≤ min[ξ 2 (α − α )dim L , ξ( ρ−ρ β (α − α ) (4.9) ρ 3µ3 −8λα , ξ =: (C βµ2 ) e then, for any integer 1 1 [log((ν − 8λ)(α − α )) − log log( )], 2 log 2 ε

1 ≤ n ≤ there exist

D ∈ N F(α , β , γ , λ, µ, ν, ρ ; Ω, P )

and covariant matrices U , V , F and W such that V (D + F )U = D + F

and

V (I + W )U = I + W


α < α,

γ =

2 3µ3 +4µ2 ( C1 βµ γ, ρ )

β = (1 + ξε )β

ρ < ρ;

(4.10)

(ii) for all k ≥ 0, | (U − I)ba |C k + | (V − I)ba |C k + | (D − D)ba |C k ≤

ε − α |b−a| k e 2 (γ ) , ξ (4.11)


| (F )ba |C k + | (W )ba |C k ≤ ε e−α |b−a| (γ )k n ε = ( ξ2 (α−αε )dim L )2 ; (iv) F and W are smooth on P = Rn (P), where

R(P) = P(5λ)δ (6λ + 2ν) ρ 2µ 1 δ = (C βµ 2) γ.

(4.12)

(4.13)

Proof. We can assume that 2α > α. Let α = α1 > α2 > . . . and ρ = ρ1 > ρ2 > . . . with αi − αi+1 = 2−i (α − α ),

ρi − ρi+1 = 2−i (ρ − ρ ),

24

L. H. Eliasson

and let εi+1 = max[

ε2i , εi e−(ν−8λ)(αi −αi+1 ) ] ξ 2 (αi − αi+1 )dim L ε2i = 2 ξ (αi − αi+1 )dim L

i ≤ n.

If we apply n times Proposition 3 we get the statement (i-iv), but only with 1 βµ2 (3µ3 +4µ2 )n ) γ. γ = ( C ρ In order to get better smoothness we must look more closely. Let us define

Vj . . . V1 (D + F )U1 . . . Uj = Dj+1 + Fj+1 Vj . . . V1 (I + W )U1 . . . Uj = I + Wj+1 . with 1 ˆ Uj = I + Kj − W j, 2

1 ˆ Vj = I − K j − W j 2

ˆj Dj+1 = Dj + G

and

1 ˆ ˆ [Kj , Dj ] + (Dj+1 − Dj ) = Fˆj − (W j Dj + Dj Wj ), 2 where ˆ denotes truncation at distance ν − 8λ from the diagonal and D1 = D and F1 = F . We now conjugate these equations to the left and right by Q−1 and Q. This ˜ j and give the equations conjugation will block diagonalize all the Dj ’s to D ˜j, D ˜j] + G ˜ j = Q−1 Fˆj Q − 1 (Q−1 W ˜j + D ˜ j Q−1 W ˜ ˆ j QD ˆ j Q) = RHS. [K 2 These equations now split into blocks: ˜ j )ba = 0 (K

and

b ˜ j )b = (RHS) ˜ (G a a

˜ a and D ˜ b differ by ≤ ρ; when any two eigenvalues of the blocks D a b b ˜ j )b (D ˜ j )b − (D ˜ j )a (K ˜ j )b = (RHS) ˜ (K a b a a a

and

˜ j )b = 0, (G a

˜ b differ by ≥ ρ. That ˜ a and D when at least two eigenvalues of the blocks D a b the Kj ’s, Dj ’s and Fj ’s are γ -smooth now follows by an easy induction. When D, F and W are Hermitian these results can be improved. In Lemma 2, G is Hermitian and K is anti-Hermitian – this is an immediate consequence of the construction we have made; the estimate in (4.2) becomes

4λα −α|b−a| k | Kab |C k + | Gba |C k ≤ constε βµ e (γ ) ρ e 2

2µ+1 γ = (const βµ γ; ρ )


25

ρ the estimate for δ in (4.3) becomes (const βγµ 2 ). In Proposition 3 and Corollary 4 D , F and W are Hermitian and V = U ∗ ; the smallness assumptions (4.4) and (4.9) are given by

ε ≤ ξ(

ρ − ρ (α − α )dim L ), β

ξ = (C

and ε ≤ min[ξ 2 (α − α )dim L , ξ(

ρ )e−6λα βµ2

ρ − ρ (α − α )dim L )] β

respectively; the estimate for γ in (4.5) and (4.10) becomes γ = (

βµ2 2µ+1 ) γ; Cρ

ρ the estimate of δ in (4.8) and (4.12) becomes (C βγµ 2 ). In the multi-level case the proof goes through in the same way and all the results remain the same with obvious modifications.

5

Block clustering

The result in Corollary 4 depends on ν – it is better the larger ν is. In general we cannot increase ν, but it may happen that the Ω-blocks of D ∈ N F(α, β, γ, λ, µ, ν, ρ; E, Ω) cluster into bigger blocks with a better separation property. Indeed it may happen that there are unions of blocks Ω = ∪i Ωai – the union depends on x – such that | E b (x) − E c (x) | ≤ ρ =⇒ (Ω )b (x) = (Ω )c (x) or dist((Ω )b (x), (Ω )c (x)) ≥ ν .

(5.1)

Of course Ω (x) is likely to be bigger so Ω (x) ⊂ {a :| a |≤ λ }

and

#Ω ≤ µ .

(5.2)

Since D is on normal form and therefore satisfies (2.5) it is natural to expect that dist(Ωai (x), Ωaj (x)) ≥ ν

(5.3)

for all smaller blocks Ωai building up Ω . Definition. The Ω-blocks of D ∈ N F(α, β, γ, λ, µ, ν, ρ; E, Ω) are said to be C(λ , µ , ν ) − clustering into Ω − blocks if (5.1-3) hold.

26

L. H. Eliasson

Then we get a better result. Proposition 5. Let D ∈ N F(α, . . . , ρ; Ω, P) be such that the Ω-blocks are C(λ , µ , ν )-clustering into Ω -blocks. Let F and W be covariant matrices, both smooth on P and satisfying | Fab |C k + | Wab |C k ≤ εe−α|b−a| γ k

k ≥ 0.

Then there exists a constant C – depending only on the dimensions of X and L – such that if

dim L µ ε ≤ min[ξ 4 (α − α )2 dim L , ξ( ρ−ρ ) ], β (α − α ) (5.4) ρ 3µ3 −8λα , ξ =: (C βµ2 ) e then there exist D ∈ N F(α , β , γ , λ , µ , ν , ρ ; Ω , P ), truncated at distance ν from the diagonal, and covariant matrices U , V , F and W such that V (D + F )U = D + F

and

V (I + W )U = I + W


α < α,

γ =

β = (1 +

2 7µ3 ( C1 βµ γ, ρ )

√

ε)β

ρ < ρ;

(ii) for all k ≥ 0 | (U − I)ba |C k + | (V − I)ba |C k + | (D − D)ba |C k ≤

(5.5)

√ − α |b−a| k εe 2 (γ ) (5.6)


| (F )ba |C k + | (W )ba |C k ≤ ε e−α |b−a| (γ )k √ 1 ε = e− 2 (ν −8λ)(α−α ) ;

(5.7)

(iv) F and W are smooth on P = Rn (P), where

R(P) = P(5λ)δ (6λ + 2ν ),

1 1 n = 2 log 2 [log((ν − 8λ)(α − α )) − log log( ε )] ρ 2µ 1 δ = (C βµ2 ) γ .

(5.8)


27

Proof. The assumption implies that D ∈ N F(α, β, γ, λ , µ , ν , ρ; Ω , P). Applying n times Corollary 4 would give the result, but with a smallness condition (5.4) that depends on λ , µ . But the occurrence of the block dimensions and block extensions enters through the block splitting and this only depends on λ, µ since D ∈ N F(α, β, γ, λ, µ, ν, ρ; Ω, P). The block dimension µ enters however in the estimate of | E −E | which is reflected in the second smallness condition in (5.4). Though Proposition 5 is not a corollary of Corollary 4, the proof is word by word the same as for Corollary 4. 5.1

Choice of parameters

How good must this clustering property be in order for us to apply Proposition 5 iteratively? We want sequences αj , βj , γj , λj , µj , νj , ρj , εj satisfying for all j ≥ 1  √   βj+1 = (1 + 2εj )β3j βj µ (5.9) γj+1 = ( C1 ρj j )7µj γj √   ε − 12 (νj+1 −8λj )(αj −αj+1 ) =e j+1

   εj ≤   ε ≤ j

3

min[(C βj µj 2 )12µj e−32λj αj (αj − αj+1 )2 dim L , ρ

j

ρj −ρj+1 (αj − βj −2j 2 dim L e αj ,

(

αj+1 )dim L )2µj+1 ]

(5.10)

where C is the constant of Proposition 5. If we assume that εj ≤ e−j , then βj ≤ constβ1 for all j, and it suffices that εj ≤ min[(

Cρj 36µ3j −96λj αj ) ,e , (αj − αj+1 )6 dim Lµj+1 , β1 µ2j ρj − ρj+1 4µj+1 −4j ) , e ]. ( 2β1

How shall we choose the sequences αj , λj , µj , νj , ρj so that (5.10) is fulfilled for all j ≥ 1. In order to simplify the discussion we assume a weak decay condition on the αj ’s and the ρj ’s: αj − αj+1 ≥ 4−j αj

and ρj − ρj+1 ≥ 4−j ρj

∀j ≥ 1.

These requirements leads to the conditions √ √ νj+1 αj νj+1 αj 4j − 1 ) ≤ αj+1 ≤ min[ j αj , j+6 exp(− j+6 ] 2 µj+2 dim L 4 2 λj+1

(5.11)

28

L. H. Eliasson

and

√

νj+1 αj ρj+1 4j − 1 ρj ) ≤ ≤ β1 4j β1 2j+6 µj+2

(5.12)

  log( α1j ) log(λj+1 )µj+2    log( β1 )µ j+2 ≥ 2j+6 dim L ρj  λj αj    2β1 µ2  log( Cj+1 )µ3j+1 .

(5.13)

exp(− and √

νj+1 αj

where the first two conditions make (5.11-12) possible. Hence, if the C(λj+1 , µj+1 , νj+1 )-clustering satisfies (5.13), then we can choose αj+1 and ρj+1 according to (5.11-12), and (5.10) will hold for all j ≥ 1 if it holds for j = 1. Notice that in the Hermitian case µj+2 can be replaced by µj+1 in (5.11-13). Proposition 6. Let D1 ∈ N F(α1 , . . . , ρ1 ; E1 , Ω1 , P1 ) and let W1 and F1 be covariant matrices, smooth on the P1 , with W1 = 0 and | (F1 )ba |C k ≤ ε1 e−α1 |b−a| γ1k

∀k ≥ 0.

Then there exists a constant C – depending only on the dimensions of X and L – such that if the sequences αj , βj , γj , λj , µj , νj , ρj , εj satisfy (5.9-10) then the following hold. (I) If the Ω1 -blocks are C(λ2 , µ2 , ν2 )-clustering into Ω2 -blocks, then there is an J ≥ 2 such that the statement (SJ ) holds: for all 1 ≤ j < J there exist Dj+1 ∈ N F(αj+1 , . . . , ρj+1 ; Ej+1 , Ωj+1 , Pj+1 ), truncated at distance νj+1 from the diagonal, and matrices Uj , Vj , Vj (Dj + Fj )Uj = Dj+1 + Fj+1 ,

Vj (I + Wj )Uj = I + Wj+1

that satisfy (5.6 − 8)j . (II) If the ΩJ -blocks are C(λJ+1 , µJ+1 , νJ+1 )-clustering into ΩJ+1 -blocks, then also (SJ+1 ) holds. (III) If (S∞ ) holds, then there exists a matrix U such that U (x)−1 (D(x) + F (x))U (x) = D∞ (x),

∀x ∈ X,

and D∞ (x) is a norm limit of Dj (x). If D and F are Hermitian, then U is unitary. (IV) Let ηj be the measure of the set of all x ∈ X such that

If

#{| a |≤ νj+1 + 2λj :| Eja (x) − Ej (x) |≤ ρj } > 1. ηj < ∞, then D∞ (x) is pure point for a.e. x.


29

Proof. (I-III) are direct consequences of Proposition 5. We get U and U −1 as the norm limit of the compositions U1 U2 . . . Uj If

and Vj . . . V2 V1 .

ηj < ∞, it follows that for a.e x

#{| a |≤ νj+1 + 2λj :| Eja (x) − Ej (x) |≤ ρj } = 1,

∀j ≥ j(x).

This implies that the blocks stop increasing, i.e. Ωj (x) = Ωj(x) (x), for a.e. x ∈ X. This proves (IV).

∀j ≥ j(x),

In the multi-level case the proofs goes through in the same way and all the results remain the same with obvious modifications. We can describe this perturbative process with a diagram. ε1 ∼ F1 + D1 ∈ N F(α1 , . . . , ρ1 ; Ω1 , P1 ) ⇓ ⇓

⇐⇐⇐

C(λ2 , µ2 , ν2 ) − clustering

⇓ ε2 ∼ F2 + D2 ∈ N F(α2 , . . . , ρ2 ; Ω2 , P2 ) ⇓ ⇓

⇐⇐⇐


⇓ ε3 ∼ F3 + D3 ∈ N F(α3 , . . . , ρ3 ; Ω3 , P3 ) ⇓ ⇓

⇐⇐⇐


⇓ Figure 1

6

Transversality of resultants

In this section we shall derive some estimates of the resultants of the matrices Dj constructed in Proposition 6, which will prove essential in showing the clustering property of blocks.

30

L. H. Eliasson

The resultant Res(D1 , D2 ) of two square matrices D1 and D2 of dimension m1 and m2 , respectively, is the product of the differences of the eigenvalues, i.e. (ei − fj ), where ei and fj are the eigenvalues of D1 and D2 respectively. All the eigenvalues are counted, so this is always a product of m1 m2 many factors. The resultant measures the difference between the spectra of the two matrices. For example | e1 − f1 | ≤ ε =⇒| Res(D1 , D2 ) | ≤ ε(| D1 ) | + | D2 |)m1 m2 −1 . The resultant Res(D1 , D2 ) = det χD2 (D1 ), where χD2 (t) = det(D2 − tI) is the characteristic polynomial of the matrix D2 . χD2 (t) is a sum of ≤ m2 ! many monomials of degree m2 in the matrix elements of D2 − tI. Introducing D1 for t we get an m1 × m1 -matrix whose elements are ≤ m2 !(m1 )m2 −1 many monomials of degree m2 in the matrix elements of these two matrices and in their differences. Taking the determinant gives this number to the power m1 , times m1 !. This shows that Res(D1 , D2 ) is a sum of less than (m1 m2 )m1 m2 many monomials of degree less than m1 m2 in the components of D1 and D2 and their differences. Moreover Res(D1 , D2 ) = det(D1 − fj I). ˆ 2 , then Res(D1 , D2 ) vanishes to the This formula shows that if D1 = D2 + εD order m1 = m2 in the variable ε. For a given normal form matrix D ∈ N F(α, . . . , ρ; Ω, P) consider, for any a, b ∈ L, the resolvents ua,b (x, y) = Res(DΩa (x+y) (x + y), DΩb (x) (x)). Since D is smooth on P we get immediately that ua,b is smooth when x, x + y vary over the pieces of the partition Ta−1 (P(λ)) ∨ Tb−1 (P(λ)) and that 2

| ua,b |C k ≤ (4dim X µ2 β)µ γ k

∀k ≥ 0.

(6.1)

Since ua,a (x, y) vanishes to order #Ωa in y we get for ua,a (x, y) as a function of x, with y fixed, (k + µ)! 2 k ) γ ∀k ≥ 0. (6.2) k! Consider now the conditions: for all x, y ∈ X and for all i a

2

| ua,a ( , y) |C k ≤ | y |#Ω γ µ (4dim X µ2 β)µ (

1 ∂ k ua,b (x, y) | ≥ σ, (k!)2 γ k yi

(6.3)

a b 1 ∂xki ua,b (x, y) | ≥ σ | y |#(Ω (x+y)∩Ω (x)) , 2 k (k!) γ

(6.4)

max |

0≤k≤J

max |

0≤k≤J

where J = #Ωa (x + y) × #Ωb (x)s and s is a fixed parameter.


31

Definition. We say that D ∈ N F(α, β, γ, λ, µ, ν, ρ; Ω) is (σ, s)-transversal on the Ω-blocks, denoted D ∈ N F(α, β, γ, λ, µ, ν, ρ; Ω)&T (σ, s), if the functions ua,b satisfy (6.3-4) for all x, y such that Ωa (x + y) and Ωb (x) are disjoint or equal. Lemma 7. Assume that D ∈ N F(α, β, γ, λ, µ, ν, ρ; Ω, P)&T (σ, s) is truncated at distance ν from the diagonal, and assume also that the Ω-blocks are C(λ , µ , ν )-clustering into Ω -blocks. Let D ∈ N F(α , β , γ , λ , µ , ν , ρ ; Ω , P ) be a perturbation of D | (D − D)ba |C k ≤

√

εe− 2 |b−a| (γ )k α

∀k ≥ 0.

If 4

ε ≤ C (µ ) ( then

γ 2s(µ )2 σ 4s(µ )4 ) ( ) γ β

(6.5)

D ∈ N F(α , β , γ , λ , µ , ν , ρ ; Ω , P )&T (σ , s)

for any 4

σ ≤ C (µ ) (

γ s(µ )2 σ s(µ )4 ) ( ) . γ β

(6.6)

The constant C depends only on the dimensions of L and X, and on s. Proof. Consider first the case (6.3). Let u ˜c,d (x, y) = Res(D(Ω )c (x+y) (x + y), D(Ω )d (x) (x)) where

Ω c = ∪Ωa ,

(Ω )d = ∪Ωb ,

are unions of less than µ many Ωa ’s which are separated by a distance at least ν. Since D is truncated at distance ν from the diagonal we get that u ˜c,d (x, y) = ua,b . a,b

Then by Lemma B2 max

0≤k≤s(µ

)2

|

1 (k!)2 (γ )k

2 σ 4 1 2 4 γ ∂yki u ˜c,d (x, y) | ≥ ( )8s (µ ) ( )s(µ ) ( )s(µ ) . (6.3) 2 γ β

32

L. H. Eliasson

Let now uc,d (x, y) = Res(D(Ω )c (x+y) (x + y), D(Ω )d (x) (x)).

Then uc,d (x, ) |C k ≤ | uc,d (x, )−˜

√

2

ε(4dim X (µ )2 )(µ )

+1

√ 2 (β+ ε)(µ ) (γ )k

∀k ≥ 0

which gives the result. The case (6.4), when the blocks are disjoint, is proven in the same way. In order to see (6.4) when the blocks are equal we consider va,b (x, y) = ua,b (x, y) when Ωa (x + y) ∩ Ωb (x) = ∅ and va,b (x, y) = ua,b (x, y)

1 |y

|#Ωa (x+y)

,

when Ωa (x + y) = Ωb (x). We define v˜c,d and vc,d in the same way as u ˜c,d and uc,d using v instead of u. We now apply the same argument to the v:s as before to the u’s. 6.1


We want sequences αj , βj , γj , λj , µj , νj , ρj , σj , εj satisfying for all j ≥ 1  √  βj+1 = (1 + εj )βj    2  γj+1 = ( 1 βj µj )7µ3j γj C √ ρj (6.7)  εj+1 = e− νj+1 −8λj )(αj −αj+1 )    4 3 2 4  σj+1 = C µj+1 (C ρj 2 )14sµj µj+1 ( σj )sµj+1 , βj βj µ j

and, besides (6.8)=(5.10), 4

εj ≤ C µj+1 (

γj 4sµ2j+1 σj 2sµ4j+1 ) ( ) , γj+1 βj

(6.9)

where C is the smallest of the constant in Proposition 5 and Lemma 7. How shall we choose the sequences λj , µj , νj , αj , ρj so that the two conditions (6.8-9) are fulfilled? If we assume that µj+1 ≥ 4sµj it follows that σj+1 ≤ (

∀j ≥ 1

Cσ1 ρj (µ1 ...µj+1 )5 ) . β1

Then we are lead to (6.10)=(5.11), √ νj+1 αj ρj+1 4j − 1 ρj ) ≤ ≤ exp(− j+6 2 (µ1 . . . µj+2 )5 β1 4j β1

(6.11)


and

√

νj+1 αj

  log( α1j ) log(λj+1 )µj+2    log( 1 )(µ1 . . . µj+2 )5 ≥ (2s)j+6 (dim L) ρj  λj αj    β1 µ2  log( Cσj+1 )(µ1 . . . µj+2 )5 . 1

33

(6.12)

The clustering required by (6.12) is much stronger than (5.13) but it will be fulfilled in the applications we have in mind. Proposition 8. Let D1 ∈ N F(α1 , . . . , ρ1 ; E1 , Ω1 , P1 )&T (σ1 , s) and assume that the D1 is truncated at distance ν1 from the diagonal. Let W1 and F1 be covariant matrices, smooth on P1 with W1 = 0 and | (F1 )ba |C k ≤ εγ1k

∀k ≥ 0.

Then there exists a constant C – depending only on the dimensions of X and L and on s – such that if the sequences αj , βj , γj , λj , µj , νj , ρj , σj , εj satisfy (6.7-9) then the following hold. (I) If the Ω1 -blocks are C(λ2 , µ2 , ν2 )-clustering into Ω2 -blocks, then there is an J ≥ 2 such that the statement (SJ ) holds: for all 1 ≤ j < J there exist Dj+1 ∈ N F(αj+1 , . . . , ρj+1 ; Ej+1 , Ωj+1 , Pj+1 )&T (σj+1 , s) and matrices Uj , Vj , Vj (Dj + Fj )Uj = Dj+1 + Fj+1 ,

Vj (I + Wj )Uj = I + Wj+1

that satisfy (5.6 − 8)j . (II) If the ΩJ -blocks are C(λJ+1 , µJ+1 , νJ+1 )-clustering into ΩJ+1 -blocks, then also (SJ+1 ) holds. (III) If (S∞ ) holds, then there exists a transformation U such that U (x)−1 (D(x) + F (x))U (x) = D∞ (x),

∀x ∈ X,

and D∞ (x) is a norm limit of Dj (x). If D and F are Hermitian, then U is unitary. (IV) Let ηj be the measure of all x ∈ X such that

If

#{| a |≤ νj+1 + 2λj :| Eja (x) − Ej (x) |≤ ρj } > 1. ηj < ∞, then D∞ (x) is pure point for a.e. x.

Proof. Immediate consequence of Proposition 6 and Lemma 7.

In the multi-level case the transversality condition applies to the resultants u(a,i),(b,j) (x, y) = Res(DΩa,i (x+y) (x + y), DΩb,j (x) (x)).

34

L. H. Eliasson

The proofs goes through in the same way and all the results remain the same with obvious modifications. We can describe this perturbative process with the diagram in Figure 2. In the next section we shall consider only matrices over T with a Diophantine quasi-periodic group action, and then we shall see that transversality on blocks implies clustering of blocks and the diagram will then be closed. ε1 ∼ F1 + D1 ∈ N F(α1 , . . . , ρ1 ; Ω1 , P1 )

&

⇓ ⇓

⇓ ⇐⇐⇐


⇓ ⇒⇒⇒ &

⇓

T (σ2 , s) ⇓

⇐⇐⇐


⇓

⇓ ⇓

ε3 ∼ F3 + D3 ∈ N F(α3 , . . . , ρ3 ; Ω3 , P3 )

⇒⇒⇒ &

⇓ ⇓

⇓ ⇓

ε2 ∼ F2 + D2 ∈ N F(α2 , . . . , ρ2 ; Ω2 , P2 ) ⇓

T (σ1 , s)

T (σ3 , s) ⇓

⇐⇐⇐


⇓

⇓ ⇓

Figure 2

7

A Perturbation theorem

In this section we shall restrict the discussion to the circle X = T and the L-action

T :L×T→T (7.1) (a, x) → (x+ < a, ω >) where ω is a Diophantine vector < a, ω > ≥

κ | a |τ

∀a ∈ L \ {0}

(7.2)

– τ > dim L and κ > 0 are fixed numbers. The numbers τ, κ will appear in the estimates but since they will be fixed throughout the iteration we shall not make them explicit in our notations. We shall use the Diophantine property in Lemma 9 and also in Lemma 11 where we shall make a particular choice of the partitions Pj . In Lemma 10 we make use of the first transversality condition (6.3), and in the proof of Theorem 12 we also use the second transversality condition (6.4).


35

Definition. We say that the eigenvalues of D ∈ N F(α, . . . , ρ; E)&T (σ, s) have almost-multiplicity ≤ µµ if, for all x and for all t ≤ ρ, the inequality | E(x + y) − E(x) |> t( is fulfilled outside at most

µ µ

µ ) µ

many intervals of length less than 1 t 1 2(4βµ ) s ( ) sµ2 . σ

Lemma 9. Let the eigenvalues of D ∈ N F(α, β, γ, λ, µ, ν, ρ; E, Ω, P)&T (σ, s) have almost-multiplicity ≤

µ µ

and assume

ρ ≤ ( Then

κ 1 2τ sµ2 ) σ. 4βµ λ

(7.3)

D ∈ N F(α, β, γ, λ, µ, ν, ρ; E, Ω, P)&C(λ , µ , ν )

for any ν =

1 µ κ 1 σ τ sµ ( )τ ( ) 2 , 2µ 4βµ ρ

λ =

Proof. For each x, | E(x + y) − E(x) |> ρ( is fulfilled outside at most

µ µ

µ (ν + 2λ). µ

(7.4)

µ ) µ

many intervals of length less than

1 1 ρ L = 2(4βµ ) s ( ) sµ2 . σ

If < a, ω >≤ L it follows that κ 1 | a | ≥ ( ) τ = S. L This means that on distances of size less than S there are at most µ µ

µ µ

many

many resonant blocks. ρ-almost resonant eigenvalues and, hence, at most Each resonant block extends over 2λ sites so there must be a gap of size at µ least µµ (S − 2λ µµ ) ≥ 2µ between the resonant blocks. And unless S = ν there is a gap of size ν between the resonant blocks, they will extend over distances which are at most λ = µµ (ν + 2λ). This proves the lemma.

36

L. H. Eliasson

We must now take into account the partition P of T. We shall for this discussion fix an integer p and define | P | to be the minimum of the length of the pieces, with the exception of the p − 1 smallest pieces. Hence, all pieces of P with the exception of at most p − 1 pieces have a length larger than this number. Lemma 10. Let D ∈ N F(α, β, γ, λ, µ, ν, ρ; E, Ω, P)&T (σ, s). (i) Then the eigenvalues of D have almost-multiplicity ≤

µ µ

with

2 2 γ µ = 2sµ 16 (4βµ2 )µ (sµ2 )2 (p+ | P(λ) |−1 ). µ σ

(ii) Assume that the Ω-blocks are C(λ , µ , ν )-clustering into Ω -blocks, and let

D ∈ N F(α , β , γ , λ , µ , ν , ρ ; E , Ω , P )&T (σ , s)

be a perturbation of D, | (D − D )ba |C 0 ≤

√

εe−α |b−a| .

If ε ≤ min[(

2 1 8sµ2 (µ )3 σ 4sµ2 µ ) ( ) , | P (λ ) |4sµ µ ] 2sβ µ γ

(7.5)

and 1

ρ ≤ ε µ ,

(7.6)

then the eigenvalues of D have almost-multiplicity ≤

µ µ

with

2

µ = p22s(µ ) .

(7.7)

Proof. Let u(x, y) be the resultant of DΩ(x+y) (x + y) and DΩ(x) (x). u(x, y) is smooth for y ∈ I, x + I ∈ P(λ), and satisfies ˜ k = (4βµ2 )µ2 γ k | u(x, ) |C k ≤ β(γ)

∀k ≥ 0.

The eigenvalues of D are bounded by βµ, so if | E(x + y) − E(x) | ≤ t( it follows that | u(x, y) | ≤ t(

µ ), µ

2 µ )(2βµ)µ . µ

t ≤ ρ,


37

We can now apply Lemma B1 to each interval I. It follows that, for each x and each t ≤ ρ µ | E(x + y) − E(x) |> t( ) µ µ µ

is fulfilled outside at most

many intervals of length less than

1 t 1 Lt = 2(4βµ ) s ( ) sµ2 , σ

i.e. the eigenvalues of D have almost-multiplicity ≤ µµ . In order to prove (ii) notice that the eigenvalues of DΩ and DΩ differ by 1 at most 2β µ ε 2µ . Using (i) it follows that | E (x + y) − E (x) | > is fulfilled outside at most

µ µ

t µ ( ) 2 µ

many intervals of length less than Lt if we let 1

t = 4β µε 2µ . By the second part of (7.5), each such interval is cut by the partition P (λ ) into at most p subintervals I. By (7.6), ρ ( µµ ) ≤ 2t ( µµ ) so we can restrict to one of these intervals I. Let u (x, y) be the resultant of DΩ (x+y) (x + y) and DΩ (x) (x). u (x, y) is smooth for y ∈ I and satisfies 2 | u (x, ) |C k ≤ β˜ (γ )k = (4β (µ )2 )(µ ) (γ )k

∀k ≥ 0.

The eigenvalues of D are bounded by β µ , so if | E (x + y) − E (x) | ≤ t (

µ ), µ

t ≤ ρ ,

it follows that

2 µ )(2β µ )(µ ) . µ By Lemma B1 this defines a union of at most

| u (x, y) | ≤ t (

γ ˜ 8β (s(µ )2 + 1)2 | I | +1) σ many intervals, each of which has length at most 2

2s(µ ) (

1 1 t 2(4β µ ) s ( ) s(µ )2 . σ By the first condition in (7.5) the number of intervals is less than

1 2s(µ )2 2 µ

38

L. H. Eliasson

7.1


We want sequences αj , βj , γj , λj , µj , νj , ρj , σj , εj satisfying for all j ≥ 1  √  βj+1 = (1 + εj )βj   2   1 βj µj 7µ3j  γ = ( γj  j+1 C ρj√ )    1  − (ν −8λj )(αj −αj+1 ) j+1 2   εj+1 = e 4 3 2 4 ρ σ σj+1 = C µj+1 (C βj µj 2 )14sµj µj+1 ( βjj )sµj+1 j   µ  λj+1 = ( µj+1 )(νj+1 + 2λj )   j  2  2sµ2j  µj+1 = p˜2 , p˜ = 16 σγ11 (4β1 µ21 )µ1 (sµ21 )2 (p+ | P1 (λ1 ) |−1 )   1   2 1 σ µj  νj+1 = 2µj+1 ( 8βj µκj+1 ) τ ( ρjj ) 4τ sµj ,

(7.8)

where C is the smallest of the constants in Proposition 5 and Lemma 7. The question is now to choose αj and ρj so that, besides (7.9)=(6.8) and (7.10)=(6.9), also  3 3 εj ≤ min[( 2sβ 1 µ σγj+1 )4sµj+1 , | Pj+1 (λj+1 ) |4sµj+1 ] j+1 j+1 j+1 1 (7.11) 1 2τ sµ2j ρj ≤ min[ε µj , ( κ σj ]. j−1 8µj+1 βj λj ) Now λj increases at least as fast as 2j , so if we define αj =

1 λj

(7.12) 1 µ

j , ρj will also decay then αj will decay exponentially fast. Since ρj ≤ εj−1 fast. Then the conditions are fulfilled if ρj satisfy  6 εj ≤ min[( Cσ1 )(µ1 ...µj+1 )6 , ρ(µ1 ...µj+1 ) , | Pj+1 (λj+1 ) |4sµ3j+1 ] j β1 1 ρ ≤ ε µj .

j

j−1

If we let (µ1 ...µj+1 )6

ρj

= εj ,

(7.13)

which defines ρj inductively, then these conditions amounts to 1

ρj ≤ min[(

4sµ3

j+1 Cσ1 (µ1 ...µj )6 µj 6 ), ρj−1 , | Pj+1 (λj+1 ) | (µ1 ...µj+1 ) ]. β1

(7.14)

The first two conditions are easily seen to be fulfilled because ρj decays superexponentially. We now must discuss the partition and Pj+1 (λj+1 ).


39

Lemma 11. We can choose the partition P in Proposition 5 in such a way that the endpoints of the pieces of P (N ), for any N are contained in ∪{Ta (endpoints of P) :| a |≤ (11λ + 2ν + where δ = (C

1 1 τ +1 ( ) )n + N }, κ δ

ρ 2µ 1 ) βµ2 γ

and

1 1 (log((ν − 8λ)(α − α )) − log log( )). 2 log 2 ε (When P has only one piece we can let an arbitrary point be its “endpoint”.) n =

Proof. Due to the Diophantine condition (7.2) any interval of length δ will contain at least one element of the orbit {Ta x :| a |≤ κ1 ( 1δ )τ +1 } for any x, in particular for x ∈ P. Hence we can choose such points as endpoints of the partition into pieces of length δ. Since P = Rn P,

RP = P(5λ)δ (6λ + 2ν ),

the result follows. It follows that the partitions from Proposition 8 can be chosen so that the endpoints of Pj (λj ) are contained in ∪{Ta (endpoints of P1 ) :| a |≤ Jj } where Jj = λ j +

j

(11λi−1 + 2νi +

i=2

2 1 1 τ +1 1 ( ) )ni ≤ ( )4(τ +1)µj−1 , κ δi−1 Bρj−1

1 1 (log((νi − 8λi−1 )(αi−1 − αi )) − log log( )), 2 log 2 εi−1 where B = B(κ, τ, s, dim L, β1 , γ1 , µ1 ). This implies that any interval of length κ ≤ τ Jj ni =

intersects at most p = #P1 many pieces of the partition Pj (λj ), i.e. | Pj (λj ) | ≥

2 κ ≥ (κBρj−1 )4(τ +1)µj−1 . τ Jj

The third condition on ρj in (7.14) therefore becomes ρj ≤ (κBρj )

3 4τ (τ +1)µ2 j 4sµj+1 (µ1 ...µj+1 )6

which is fulfilled if for example µ21 ≥ 16sτ (τ + 1). We can now derive the following theorem.

40

L. H. Eliasson

Theorem 12. Let D ∈ N F(α, . . . , ρ; Ω, P)&T (σ, s) be covariant with respect to the quasi-periodic L-action (7.1-2), and assume that D is truncated at distance ν from the diagonal. Let F be a covariant matrix, smooth on P and | Fab |C k ≤ εe−α|b−a| γ k ∀k ≥ 0. Then there exists a constant C – C depends only on dim L, κ, τ, s, α, β, γ, λ, µ, ν, ρ, σ, #P – such if ε ≤ C then there exists a matrix U such that U (x)−1 (D(x) + F (x))U (x) = D∞ (x),

∀x ∈ X,

and D∞ (x) is a norm limit of normal form matrices Dj (x). Moreover D∞ (x) is pure point for a.e. x. The limit limj→∞ Ej (x) = E∞ (x) is uniform and satisfies, for all y ∈ / 2πZ, Lebesgue{x : E∞ (x + y) − E∞ (x) = 0} = 0. Moreover, if the Ej ’s and E∞ are real, then for all subsets Y −1 (Y ) = 0 Lebesgue(E∞

if

Lebesgue(Y ) = 0.

If D and F are Hermitian, then U is unitary and D∞ is Hermitian. Proof. Define the sequences αj , βj , γj , λj , µj , νj , ρj , σj , εj by the formulas (7.8) and (7.12-13), starting with α, β, γ, λ, µ, ν, ρ, σ, ε. Then they verify the assumptions (6.7-9), so we can apply Proposition 8, and (7.11), so we can apply Lemma 9-11. Use now Proposition 8(I-II). By Lemma 9–11 we get C(λj , µj , νj )-clusteering for all j ≥ 1, hence statement (S ∞ ) holds. The first part of the theorem now follows from Proposition 8(III) – it also gives the Hermitian case. In order to prove that D∞ (x) is pure point for a.e. x we must estimate the number ηj in Proposition 8(IV). This is the only place where we use condition (6.4). For each a in | a |≤ νj+1 + 2λj we consider uj (x, x + aω) = Res((Dj )Ωj (x+y) (x + aω), (Dj )Ωj (x) (x)) and the set | uj (x, x + aω) | ≤

ρj µ2

.

βj j

Using Lemma B1 we get that this this is a union of intervals of length less than 2 (νj+1 + 2λj )τ sµ12j ) . (2ρj Lj = γj σj κ The number of intervals does not exceed 2

Mj = 2sµj [

2 γj (νj+1 + 2λj )τ 8(βj µ2j )µj (sµ2j + 1)2 + #P)]. κσj


41

This gives an upper bound for ηj , ηj ≤ const(νj+1 + 2λj )dim L Lj Mj and the verification that

ηj < ∞

is straight forward (with the choice we have made of νj+1 ). The limit is uniform since | Ej+i (x) − Ej (x) | ≤ | Dj+i (x) − Dj (x) | . Clearly, {x : E∞ (x+y)−E∞ (x) = 0} ⊂ {x :| Ej (x+y)−Ej (x) |≤| D∞ (x)−Dj (x) |} and this set is easy to estimate using the transversality on blocks of the matrix Dj . Suppose now that the Ej ’s and E∞ are real and let Y be a set of measure 0. We can use a covering ∪i Ii ⊃ Y with the following property: each Ii is contained in an interval I˜i , and each I˜i is contained in a piece of the partition Pji (λji ), ji ≥ J, in such a way that its distance to the boundary of I˜i is at least | D∞ (x) − Dji (x) | . Then Eji is smooth on I˜i and i

−1 Lebesgue(E∞ (Ii )) ≤

Lebesgue(Ej−1 (I˜i )). i

i

This sum is easy to estimate – it is essentially equal to its first terms because of the rapid convergence – using the estimate of the resultants. Since we can −1 (Y ) is 0 if the measure do this for any J, this shows that the measure of E∞ of Y is 0. In the multi-level case the proofs goes through in the same way and all the results remain the same with obvious modifications. We can now complete the diagram of this perturbative process in Fig. 3.

42

L. H. Eliasson

ε1 ∼ F1 + D1 ∈ N F(α1 , . . . , ρ1 ; Ω1 , P1 ) ⇓ ⇓

⇐⇐⇐

T (σ1 , s)

& ⇓

⇓


⇓

⇓

⇓

ε2 ∼ F2 + D2 ∈ N F(α2 , . . . , ρ2 ; Ω2 , P2 )

⇒⇒⇒ &

⇓ ⇓

⇐⇐⇐

T (σ2 , s) ⇓

⇓


⇓

⇓

⇓

ε3 ∼ F3 + D3 ∈ N F(α3 , . . . , ρ3 ; Ω3 , P3 )

⇒⇒⇒ &

⇓ ⇓

⇐⇐⇐

T (σ3 , s) ⇓

⇓


⇓

⇓

⇓ Figure 3

8

Applications

8.1

Discrete Schr¨ odinger Equation

We consider the equation in strong coupling −ε(un+1 + un−1 ) + V (θ + nω)un = Eun ,

n ∈ Z,

(8.1)

where V is a real valued function on the one-dimensional torus T = R/(2πZ) and ω is a real number. We assume that V is piecewise Gevrey smooth on a partition with p many pieces and | V |C k ≤ βγ k

∀k ≥ 0.

(8.2)

The functions V (θ + y) − V (θ)) satisfies for all θ, y the two transversality conditions

max0≤k≤s | ∂yk (V (θ + y) − V (θ)) | ≥ σ (8.3) max0≤k≤s | ∂θk (V (θ + y) − V (θ)) | ≥ σ y . We assume also that ω is Diophantine, i.e. for some τ > 1, κ > 0 nω ≥

κ | n |τ

∀n ∈ Z \ 0.

(8.4)

Theorem 13. Under the assumptions (8.2-4) there exist a small constant ε0 (p, β, γ, s, σ, κ, τ ) and, for all | ε |≤ ε0 , a function E∞ (θ) such that for almost all θ ∈ T and for all k ∈ Z, equation (8.1) has a solution uk ∈ l2 (Z) with E = E∞ (θ + kω). The set {uk : k ∈ Z} is an orthogonal basis for l2 (Z).


43

If we identify the right hand side of (8.1) as an operator Hθ , then the statement says that Hθ is pure point for almost all θ. The spectrum of Hθ is the closure of Range(E∞ ) and it is easy to verify that the Lebesgue measure of [inf V, sup V ] \ Range(E∞ ) goes to 0 when ε goes to 0. Proof. We identify (8.1) with a matrix D + εF as in the introduction. The matrix D is diagonal and D ∈ N F(α = 1, β, γ, λ = 1, µ = 1, ν = 1, ρ = 1). Since max

0≤k≤sµ2

|

1 σ ∂yk (V (θ + y) − V (θ)) | ≥ = σ ˜, 2 k (k!) γ (s!)2 γ s

and similar for the θ-derivative, we have also D ∈ T (˜ σ , s). The perturbation εF satisfies ∀k ≥ 0. | (εF )ba |C k ≤ (εe)e−α|a−b| The theorem is now an immediate consequence of Theorem 12.

In weak coupling we consider the equation −(un+1 + un−1 ) + εV (θ + nω)un = Eun ,

n ∈ Z,

(8.5)

where V is a real valued function on the torus Td and ω ∈ Rd . We assume that V is analytic, satisfying sup | V (θ) | ≤ 1,

(8.6)

| θ| d, κ > 0 < k, ω > ≥

κ | k |τ

∀k ∈ Zd \ 0.

(8.7)

Theorem 14. Under the assumptions (8.6-7) there exist a constant ε0 (r, κ, τ ) and for all | ε |≤ ε0 a function E∞ (ξ) such that for almost all ξ ∈ T and for all k ∈ Zd , equation (8.5) has a solution ukn = ein(ξ+) U k (θ + nω) in l∞ (Z) with E = E∞ (ξ+ < k, ω >). The set {U k : k ∈ Zd } is an orthogonal basis for L2 (Td ). Proof. The equation for U ∈ L2 (Td ) is identified, in Fourier coefficients, with a matrix D + εF as in the introduction. The matrix D is diagonal and D ∈ N F(α = r, β = 2, γ =

1 , λ = 1, µ = 1, ν = 1, ρ = 1). r

44

L. H. Eliasson

The eigenvalue is E(ξ) = cos(ξ) and the functions E(ξ + y) − E(ξ) satisfies the transversality conditions (8.3) for all ξ, y with some σ and s = 2. (Any non-constant analytic function with primitive period 2π satisfies condition (8.3) for some σ, s.) Hence D ∈ T (˜ σ , 2). The Fourier coefficients of V decays exponentially with the factor α = r which shows that εF satisfies the required smallness condition. The theorem is now an immediate consequence of Theorem 12. These two theorems claims nothing about exponential decay of the solutions uk or analyticity of the U k , i.e. exponential decay of its Fourier coefficients. The uniform exponential decay of all the uk ’s is measured in the iteration process by the sequence αj which goes to 0, and must do so. If we instead let α vary over the partition so that we had one α-value for each block, it follows from the proof that αj (x) stops decaying towards 0 as soon as the blocks Ωj (x) stops increasing. This will give exponential decay. (Exponential decay of eigenvectors have been proven in [3,4,6].) 8.2

Discrete Linear Skew-Products

We consider first the weakly perturbed skew-product of the form Xn+1 − (A + εB(θ + nω))Xn = EXn ,

n ∈ Z,

(8.8)

where A ∈ Gl(N, R), B : Td → gl(N, R). We assume that B is analytic, satisfying sup | B(θ) | ≤ 1

(8.9)

| θ|). The set of functions {Y k,j : k ∈ in l∞ (Z) ⊗ CN with E = E∞ d Z , j = 1, . . . , N } is an orthogonal family in L2 (Td ) × CN .

Proof. Assume first that A is semi-simple. Since we construct complex-valued solutions we can without restriction assume that A and B are complex valued and that A is diagonal with diagonal elements a1 , . . . , aN . The equation for Y ∈ L2 (Td ) ⊗ CN is identified, in Fourier coefficients, with a matrix D + εF as in the introduction. The matrix D is diagonal and D ∈ N F(α = r, β, γ =

1 , λ = 1, µ = 1, ν = 1, ρ = 1), r


45

with β =| A |. It is now a multi-level matrix with eigenvalues E j (ξ) = eiξ − aj ,

j = 1, . . . , N

and the blocks Ωj = {0} × {j}, j = 1, . . . , N . The functions E i (ξ + y) − E j (ξ) satisfy the transversality conditions (8.3) for all ξ, y with some σ and s = 1. Hence D ∈ T (˜ σ , 1). The Fourier coefficients of B decays exponentially with the factor α = r which shows that εF satisfies the required smallness condition. The theorem is now an immediate consequence of Theorem 12 – multi-level version. The case when A is not semi-simple is almost the same. The matrix D is still on normal form, D ∈ N F(α = r, β, γ =

1 , λ = N, µ = N, ν = 1, ρ = 1), r

with β =| A |. It is still a multi-level matrix with eigenvalues E j (ξ) = eiξ − aj ,

j = 1, . . . , N

but with the blocks Ωj = {0} × {1, . . . , N }, j = 1, . . . , N , reflecting that it is block diagonal instead of diagonal. The transversality of the functions E i (ξ +y)−E j (ξ) implies D ∈ T (˜ σ, N 2 ) through Lemma B2. Now we can apply theorem 12. The right conjecture about equation (8.8), under conditions (8.7,9), is the following: for E = 0 (or for any fixed E), if ε is small enough then (8.8) has a Floquet representation for almost all matrices A. This means that the fundamental solutions can be written as en∆ Z(θ + nω),

n ∈ Z,

where Z : Td → Gl(N, R) and ∆ ∈ gl(N, R). Theorem 15 and its proof is probably not so far from proving this conjecture. An even stronger result may be true: instead of “almost all matrices A” we may put “almost all matrices in any “generic” one-parameter family At ”. Such a result has been proven for certain one-parameter families in Sl(2, R) [13] and in compact groups [17,18]. The application to the strongly perturbed skew-product (1.4) is straight forward, but the continuous Schr¨ odinger in one dimension in strong coupling (1.5) is more delicate. Theorem 12 will not apply to the discrete equation (1.6), derived from (1.5) in the introduction, since W will not satisfy the full transversality condition. It seems however that W is transversal in a neighborhood of W −1 (0) when E is close to inf V , if V is transversal near inf V . We therefore believe that when ε small enough and when E is sufficiently close to inf V , then the spectrum of the operator (1.6) is pure point near 0 – this has been proven in some cases [4].

46

L. H. Eliasson

A

Appendix A

Gevrey classes Let uj , j = 1, 2 be smooth functions in the Gevrey class G2 , defined in a convex set V in Rd , and consider the norm | uj |C =:

sup sup | 0≤l≤k x∈V

1 l ∂ uj (x) | (l!)2

– here k and l are multi-indices in Nd . Lemma A.1 Assume that | uj |C k ≤ βj γ k , ∀k ≥ 0. Then for all k ≥ 0 (i) | u1 u2 |C k ≤ 4d (β1 β2 )γ k ; (ii) | eu1 |C k ≤ e4 (iii) |

d

β1 k

γ ;

√ 1 + u1 |C k ≤ (1 + β1 )γ k

if

4d β1 < 1;

(iv) |

1 4d β1 k 1 γ) |C k ≤ ( u1 δ δ

if

| u1 |> δ.

Proof. k ∂ k−l u1 ∂ l u2 | l 0≤l≤k k (l!)2 ((k − l)!)2 ≤ (β1 β2 )γ k (k!)2 , l (k!)2

| ∂ k (u1 u2 ) | = |

0≤l≤k

and the result follows since d ki −1 k −1 = [ ] ≤ 4d l l i i=1

0≤l≤k

0≤li ≤ki

for all k – just notice that klii ≥ 2li if li ≤ k2i . √ We use the power series expansions of eu and 1 + u which converges absolutely in | u |< ∞ and in | u |< 1 respectively, to prove the second and third estimate.


47

Notice that | u1 |> δ implies that β1 > δ. By a scaling we can suppose that δ = 1 and β1 ≥ 1. The fourth statement is obvious for k = 0 so we proceed by induction on | k |. Hence k k−l 1 l 1 k 1 ∂ ( )∂ u1 | |∂ ( )| =|− l u1 u1 u1 0≤l≤k, l=0 k (l!)2 ((k − l)!)2 d(|k|−1) 2 k ≤ 4 (k!) (β1 γ) l (k!)2 0≤l≤k

≤ (k!) (4 β1 γ) . 2

d

k

Remark. The preceding lemma is valid also for matrices if we understand by u1 the inverse of u, and if we use the operator norm satisfying | uv |≤| u || v | . Estimates of eigenvalues Let D(x) be a m × m-matrix, smooth in a convex set V in Rd containing 0, and let {E j (x)}m 1 be a continuous choice of its eigenvalues. Lemma A.2 (i) | E j (x) | ≤ | D(x) | . (ii) 1

1

| E j (x) − E j (x ) | ≤ 4 | D |C 0 m | ∂D |Cm0 | x − x | m . ¯ then (iii) If D is Hermitian, i.e. Dt = D, 1−

1

| E j (x) − E j (x ) | ≤ | ∂D(x) |C 0 | x − x | . Proof. (i) is obvious. To see (ii), let I be the segment joining x and x and let δ =| E 1 (x) − E 1 (x ) |. Then there is an x ∈ I such that, δ | P (λ, x) | ≥ ( )m , λ = E 1 (x ), 4 m j where P (λ, x) = i=1 (λ − E (x)). (This is a well-known inequality of Chebyshev – see e.g. [28].) Hence δ ( )m ≤ | P (λ, x) − P (λ, x ) | ≤ | ∂x P (λ, ) |C 0 | x − x | . 4 Since | ∂x P | ≤

m j=1

| det(D1 . . . ∂x Dj . . . Dm ) | ≤

m i,j=1

| ∂Dij || D |m−1

48

L. H. Eliasson

and | ∂Dij |≤ m | ∂D | the result follows. To see (iii) let q j (x) be the eigenvector corresponding to E j (x). Then (D(x) − E j (x)I)q j (x) = 0, and if we differentiate the relation and take the scalar product with q j (x) and use that the eigenvectors are orthogonal, then we get an estimate of ∂E j . When Ej and q j are not differentiable one uses the same argument with a difference operator. Better estimates requires separation of eigenvalues. Assume that | D |C k ≤ βγ k

∀k ≥ 0.

Lemma A.3 Assume that the eigenvalues belong two groups

E 1 (x), . . . , E n (x) E n+1 (x), . . . , E m (x) such that any eigenvalue of one group is separated from any eigenvalue of the other group by at least r for all x ∈ V . Then the polynomial n

(λ − Ej (x)) =

j=1

n

en−j (x)λj

j=0

is smooth in V and satisfies n j βm m k β ((const | ej |C k ≤ ) γ) j r

∀k ≥ 0.

If D is Hermitian, then we also have n j β jγ. | ∂ej |C 0 ≤ j The constants are independent of m. Proof. Notice that r ≥ 2β, and that the sup-estimate of the coefficients and, in the Hermitian case, of the first derivative of the coefficients follows from Lemma A.2. By scaling we can assume that β = 1 if we replace r by βr . The polynomial P (λ, x) = det(λI − D(x)) satisfies | P (λ, ) |C k ≤ ((const)m γ)k if just | λ |< 2.

∀k ≥ 0,


49

Choose a curve ∆(x) in | λ |≤ 1 + 12 – piecewise constant in x – keeping a distance ≥ 2r to all the roots E 1 (x), . . . , E m (x) of P (λ, x) and surrounding the first n of these roots. ∆(x) may consist of several components so we can choose it to be of length at most nπr. Then we have for all λ ∈ ∆(x)

| P (λ, x) | ≥ ( 2r )m = ρ (λ,x) | ∂λPP(λ,x) | ≤ 2m r . Using this we verify, as in Lemma A.1, that |

2m 1 ∂λ P (λ, x) |C k ≤ ((const)m γ)k P (λ, x) r ρ

∀k ≥ 0

– where we have used the Cauchy formula to estimate | ∂λ P (λ, ) |C k on | λ |≤ 1 + 12 . Consider now the power symmetric functions in the first n roots: pj (x) = E 1 (x)j + · · · + E n (x)j . We have the integral representation 1 ∂λ P (λ, x) dλ, λj pj (x) = 2π ∆(x) P (λ, x) from which we get 1 | pj |C k ≤ mn2j ((const)m γ)k ρ

∀k ≥ 0.

The ej ’s are the elementary symmetric functions in we have the relation  p1 1 0 0  p2 p1 2 0  (−1)j p p1 3 ej = det  3 p2 . j! .. .. .. . . . . .

the first n roots and  ...  . . .  . . . , ..   . 

pj pj−1 pj−2 pj−3 . . . for j ≥ 1 [29, page 20] – of course e0 = 1. So j!ej is a sum of at most 2j many products pι1 · · · · · pιl , ι1 + · · · + ιl = j ≤ m, each with a coefficient that is at most (j − 1) · (j − 2) · · · · · l. From Lemma A.1 we now get the estimate we want. Orthogonalization Let v 1 , . . . , v m be vectors in Cn which are smoothly defined in a convex neighborhood V of 0 in Rd . Assume that v 1 (0), . . . , v m (0) is ON and that | v j |C k ≤ βγ k

∀k ≥ 0,

for j = 1, . . . , m. (This implies in particular that β ≥ 1.)

50

L. H. Eliasson

Lemma A.4 Then there is a constant, independent of m, and an upper triangular m × n-matrix R, R(0) = I, smooth in W = {x :| x |
|C k ≤ const i = j m const i i |< v , v > −1 |C k ≤ m in | x |< const mβγ . In particular they remain linearly independent and of norm ≤ 2 if the constant is small enough. Let now m−1 vˆm = v m − ai v i i=1

and determine the ai ’s so that < vˆm , v >= 0 for all i = 1, . . . , m − 1. This gives a system of linear equation in the ai ’s which we can solve by Gauss elimination if the constant is small enough (independent of m). The solution satisfies

k | ai |C k ≤ const i = 1, . . . , m − 1 m γ m k | vˆ |C k ≤ constβγ i

for all k ≥ 0. If the constant is small enough we get | vˆm | ≥ and we can estimate wm = tion.

v ˆm |ˆ vm |

1 2

by Lemma A.1. Then we proceed by induc-

Subspaces and Angles Assume now that Λ1 and Λ2 are two subspaces in Cm with Λ1 ∩ Λ2 = {0}, and recall that they come equipped with ON-frames


51

Sublemma. The spectrum of the matrices Λ∗1 Λ2 Λ∗2 Λ1 and Λ∗2 Λ1 Λ∗1 Λ2 only differ by an eigenvalue 0 of multiplicity = | dim Λ1 − dim Λ2 |. The spectrum of I − Λ∗1 Λ2 Λ∗2 Λ1 is contained in [0, 1]. Proof. Notice that if λ = 0 is an eigenvalue of AB, then it is also an eigenvalue of BA, with the same multiplicity. Hence, the spectrum of AA∗ and A∗ A is the same except possibly for an eigenvalue 0. Now the kernel of A∗ A is precisely the kernel of A, so if A is an k × l-matrix of rank r then the dimension of the kernel is l − r. Since the rank of A and A∗ is the same, we get that 0 is an ∗ eigenvalue of AA and A∗ A of multiplicity k − r and l − r respectively. I A and Λ2 = , then If now Λ1 = 0 B Λ∗1 Λ2 Λ∗2 Λ1 = AA∗ ,

Λ∗2 Λ1 Λ∗1 Λ2 = A∗ A,

so the first first statement is proved. Since Λ∗2 Λ2 = A∗ A+B ∗ B, we know that I −A∗ A = B ∗ B. Now any matrix of the form B ∗ B is positive definite which implies that spectrum of I − A∗ A belongs to [0, ∞[ and to ] − ∞, 1]. This proves the second statement. We can now define the angle between Λ1 and Λ2 as the unique ϕ ∈ [0, π[ such that dist(σ(Λ∗1 Λ2 Λ∗2 Λ1 ), 1) = sin(ϕ). If these spaces are one-dimensional, this notion coincides with the “usual” angle between vectors. Consider now an m × m-matrix D, | D |≤ β, whose eigenvalues belongs to two groups that are separated by a distance at least r. Let Λ1 and Λ2 be the invariant subspaces corresponding to the two groups. Lemma A.5 The angle between the spaces Λ1 and Λ2 ≥ (const

1 m+1 r (m+1) m 2 , ) 2 ( ) m β

if r < constβ. Proof. We can assume that

D=

S A 0 T

,

where S and T are upper triangular with the diagonal elements of one separated by r from those of the other. We now look for an m1 × m2 -matrix R, m = m1 + m2 , m2 ≤ m1 , such that −1 I R I R S 0 D = . 0 I 0 I 0 T

52

L. H. Eliasson

I

Then the vectors

0

R

and

I

will span Λ1 and Λ2 respectively. For a matrix M denote by Mi the matrix whose (k, l)-entry is Mkl if l−k = i and 0 otherwise. Then S = S0 + · · · + Sm1 ,

T = T0 + · · · + Tm2 ,

A = A−m1 + · · · + Am2 .

The equation for R, SR − RT = −A, can now be written S0 Rj − Rj T0 = −Aj −

Sk Rj−k +

k≥1

Rj−l Tl

l≥1

for −m1 ≤ j ≤ m2 . From this we get that β | R | ≤ constm( )m+1 . r It follows from this by Gram-Schmidt that we get an ON-basis for Λ2 A B where B is a triangular m2 × m2 -matrix whose diagonal entries have absolute value 1 r ≥ const ( )m+1 , m β and | B | satisfies the same estimate as | R |. If now v ∈ Cm2 with | v |= 1, then Av 2 1 ≥| | = | Av |2 + | Bv |2 , Bv which implies that | A |2 ≤ 1 − [(const Since

1 r m+1 m2 2 ( ) ) ] . m β

| A∗ A | =| A |, it follows that | A∗ A | ≤ 1 − [(const

1 r m+1 m2 2 ( ) ) ] , m β

which bounds the spectrum of A∗ A. Since sine of the angle equals the square root of dist(σ(A∗ A), 1), the result follows.


53

Block splitting Let D be an m × m-matrix, smooth in a convex neighborhood V of 0 in Rd , and assume that | D |C k ≤ βγ k ∀k ≥ 0. Lemma A.6 For any r > 0 there exists a D(x)-invariant decomposition Cm =

k

Λj (x),

i=1

smooth on W = {| x |≤ (const

r 2m 1 ) }∩V βm γ

with the following properties: (i) all eigenvalues of DΛj are r-close, i.e. for any given x E and E are eigenvalues of the same block D(x)Λj (x) ⇒

| E − E |≤ r;

(ii) βm 2m k ) γ) r (iii) the angles between the different Λj ’s are | Λj |C k ≤ ((const

≥ (const

∀k ≥ 0;

r m2 ) ; βm

(iv) the matrix Q = (Λ1 . . . Λk ) satisfies | Q−1 |C k ≤ (const

βm 2m2 βm 4m2 k ) ) ((const γ) r r

∀k ≥ 0.

All the constants are independent of m. Proof. We can take β = 1 if we replace r by βr . Make now a decomposition r -separated groups as possible, and of the eigenvalues of D(0) into as many m consider the corresponding decomposition of Cm into invariant subspaces 1 m Λj (0). Choose a basis v (0), . . . , v (0) which is ON in each Λj (0). r By Lemma A.2 the groups of eigenvalues of D(x) remain 2m -separated for x in r m1 ) } ∩ V. U = {| x | ≤ (const βm γ ˇ (λ−E i (x)), Let {E j (x)}m j=1 be the eigenvalues of D(x) and let Pj (λ, x) = where the product is over all eigenvalues except those associated to Λj (0) – by Lemma A.3 we have estimates of the coefficients of Pˇj (λ, x).

54

L. H. Eliasson

For v i (0) in Λj (0), let 1 Pˇj (D(x), x)v i (0), i ˇ Pj (E (0), 0)

v i (x) =

x ∈ U.

This gives a basis for Λj (x) for x near 0 – how near? We have m m | v i |C k ≤ ( )m−1 (const)m ((const )m γ)k ∀k ≥ 0. r r Hence, for x ∈ W , v i (x) will be close to v i (0) and we get a basis defined in W . Lemma A.4 gives an ON-basis for each Λj (x) on W which fulfills the estimate. The estimate of the angle follows from Lemma A.5, using that m2 ≥ (m + 1) m 2. In order to estimate the inverse of Q, consider the D∗ -invariant decomposition k ˜ j (x), Cm = Λ i=1

such that the eigenvalues of D∗ (x)Λ˜ j (x) are complex conjugates of the eigen˜ j satisfies the same estimate as Λj and Λ ˜j = values of D(x)Λj (x) . Then Λ ⊥ ( i=j Λi ) . If we let ˜ = (. . . Λ ˜ iΛ ˜j . . . ) Q then

 B

−1

   ∗ ˜ =: Q Q =   

..

 .

     

˜ i )∗ Λi (Λ ˜ j )∗ Λj (Λ ..

.

˜∗. so that Q−1 = B Q ˜ j )⊥ is at least The angle between Λj and ( i=j Λi ) = (Λ (const

r m2 ) . m

Therefore ˜ j )∗ Λj = I − (Λj )∗ (Λ ˜ j )⊥ ((Λ ˜ j )⊥ )∗ Λj ˜ j (Λ Sj =: (Λj )∗ Λ is a symmetric matrix with spectrum bounded away from 0 by at least the square of that angle. Hence, by Lemma A.1, 2 2 m m | Sj−1 |C k ≤ (const )2m ((const )4m γ)k ∀k ≥ 0. r r −1 ∗ −1 ∗˜ ˜ Since ((Λj ) Λj ) = S (Λj ) Λj we get the same estimate for B as for S −1 j

and, hence, the same estimate for Q−1 as for Sj−1 .

j


55

Remark. When D is Hermitian in Lemma A.6 the angle between the invariant spaces is of course π2 and Q is unitary so | Q−1 |C k ≤ ((const

βm 2m k ) γ) r

∀k ≥ 0.

Truncation Let D : l2 (L) → l2 (L) be an infinite-dimensional matrix with a basis of finite-dimensional (generalized) eigenvectors {q b } and blocks Ωb . Suppose that D∗ also has a basis of (generalized) eigenvectors q˜b with the same blocks Ωb . We enumerate the eigenvectors so that q b and q˜b correspond to complex conjugate eigenvalues with the same multiplicity. This implies that q˜b ⊥ q a , for all a = b. Lemma A.7 Let Ω = (Ωb ) = ∪Ωa ∩Ωb Ωa ,

b ∈ L.

If v is an (generalized) eigenvector of DΩ then either v is an (generalized) b eigenvector of D or v ⊥ CΩ . ∗ Proof. Notice that q a and qã are (generalized) eigenvectors of DΩ and DΩ a b respectively, if Ω ∩ Ω = ∅. Let now v be an (generalized) eigenvector of DΩ . Then either v = q a for a some a such that Ωa ∩ Ωb = ∅ – and wec are done – or v ⊥ q˜ for all a such a b that Ω ∩ Ω = ∅. Since now v = Ac q we get

0 = < qã , v > = Aa < qã , q a >, which implies that Aa = 0. This means that v = Ac q c Ωc ∩Ωb =∅ b

so v is perpendicular to CΩ .

A numerical lemma Lemma A.8 a0 =a,a1 ,...,aj =b

√ 6 e−(|a−a1 |+···+|aj−1 −b|)s ≤ ( dim L )(j−1) dim L s

∀a, b ∈ L.

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L. H. Eliasson

Proof. If we take | · | as the l1 -norm we get e−(|a−a1 |+···+|aj−1 −b|)s = a0 =a,a1 ,...,aj =b

e−(|(b−a)−c1 −···−cj−1 |+|c1 |+···+|cj−1 |)s ≤

c1 ,...,cj−1

(

e−(| j−1 −c|+|c|) 3 )j−1 ≤ (| b−a

s

c

s 6 b−a 6 | + )(j−1)d e−|b−a| 3 ≤ ( )(j−1)d j−1 s s

(d = dim L), where we used the inequality |d−

j−1

ci | +

j−1

1

1

1 d − ci | + | ci |). (| 3 1 j−1 j−1

| ci | ≥

This inequality is easy to verify because the 3×RHS is less then d | +2 | ci |) =| d | +2 | ci |≤ j−1 |d− ci | + | ci | +2 | ci |≤| d − ci | +3 | ci |,

(|

where summation √ goes from 1 to j − 1. The factor dim L comes in because we use th l2 -norm.

B

Appendix B

Estimates of preimages Let u be a real or complex valued smooth function defined on an open interval ∆ in R and satisfying

| u |C k ≤ βγ k ∀ 0≤k ≤m+1 1 k max | (k!)2 γ k ∂ u(x) | ≥ σ ∀x ∈ ∆. 0≤k≤m

Lemma B.1 There exists a finite union of open intervals ∪j∈J ∆j such that  2  ≤ 2m [ 8βγ(m+1) | ∆ | +1] #J σ 1 2 2ρ m maxj∈J | ∆j | ≤ γ ( σ )   | u(x) | ≥ρ ∀x ∈ ∆ \ ∪∆j . Proof. Assume for simplicity that 0 ∈ ∆ and that | ∂ m u(0) |≥ σ(m!)2 γ m . ˜ of length Then there is an interval ∆ σ 8βγ(m + 1)2


57

˜ if u is real – if u is complex valued such that | ∂ m u(x) |≥ σ2 (m!)2 γ m on ∆, then this holds for u or u. ˜ There exists an interval ∆1 of length Consider now ∂ m−1 u on ∆. ≥

2 2ρ 1 ( )m γ σ

such that σ 2ρ 1 ( ) m ((m − 1)!)2 γ m−1 2 σ

| ∂ m−1 u(x) |≥

˜ \ ∆1 . ∀x ∈ ∆

˜ \ ∆1 . There exist two intervals ∆2 , ∆3 , each of Consider now ∂ m−2 u on ∆ length 2 2ρ 1 ( )m , γ σ such that | ∂ m−2 u(x) |≥

σ 2ρ 2 ( ) m ((m − 2)!)2 γ m−2 2 σ

˜ \ ∆ 1 ∪ ∆2 ∪ ∆3 , ∀x ∈ ∆

etc. ˜ 2m − 1 (possibly void) intervals ∆i such that Hence we obtain in ∆, | u(x) | ≥ ρ | ∆i | ≤

˜ \ ∪∆i , ∀x ∈ ∆ 2 2ρ 1 ( )m . γ σ

On the whole interval ∆ we get at most 2m × [ many such intervals.

8βγ(m + 1)2 | ∆ | +1] σ

Transversality of products of functions Let now uj be a sequence of real or complex valued smooth functions defined on an open interval ∆ in R and satisfying

∀ 0 ≤ k ≤ m + 1 = m1 + · · · + mj + 1 | uj |C k ≤ βγ k max | (k!)12 γ k ∂ k uj (x) | ≥ σ ∀x ∈ ∆. 0≤k≤mi

Lemma B.2 If u = u1 · · · · · uj , then

| u |C k ≤ 4d(j−1) β j γ k 2 max | (k!)12 γ k ∂ k u(x) | ≥ ( 12 )8m σ j(m+1)

0≤k≤m

∀ 0 ≤ k ≤ m + 1, ∀x ∈ ∆.

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L. H. Eliasson

Proof. The first part follows from Lemma A.1 so we concentrate on the second estimate. We can assume that β = 1 and γ = 1 if we replace the ui ’s by uβi and rescale x. Fix an x – x = 0 say. We can assume without restriction that 1 | ∂ mi ui (0) | ≥ σ (mi !)2

∀i.

Order the mi ’s in decreasing order m1 ≥ m2 ≥ . . . . Fix a symmetric interval I around 0 of length 2δ ≤

σ . 8(m + 1)2

By Lemma B1 we get that | u1 |≥ ρ outside a subset of I of measure less than 2ρ 1 N = 2m1 +2 , N ( ) m1 , σ which is less than δj if we chose ρ =

σ δ m1 ( ) . 2 jN

Applying this to all the ui ’s we get that σ δ m ) | u(x) | ≥ ρj = ( )j ( 2 jN outside a set of measure less than δ. Consider now the Taylor expansion u(x) = a0 + a1 x + · · · + am xm + u ˆ(x). Then, for | x |≤ δ, |u ˆ(x) | ≤ (m + 1)! | x |m+1 ≤ (m + 1)!δ m+1 . From this we get an

δ 2

≤| x |≤ δ such that

| a0 + a1 x + · · · + am xm | ≥ if we choose δ =

1 σ j δ m ( ) ( ) , 2 2 jN

σ 1 1 m ( )j ( ) . 2(m + 1)! 2 jN

Hence for δ so small and some | ak xk | ≥

δ 2

≤| x |≤ δ and some 0 ≤ k ≤ m, σ 1 δ m ( )j ( ) . 2(m + 1) 2 jN

Since k!ak = ∂ k u(0)the result follows.


59

Remark. In [5] there was given a short proof of this lemma with a worse estimate, which however is good enough for the applications we have in mind. The above proof, which is just as short but gives a considerable better estimate, is due to M. Goldstein [6]. Acknowledgement. This paper was partially written during a visit to IMPA, Rio de Janeiro, financed by the STINT foundation.

References 1. Craig, W.: Pure point spectrum for discrete almost periodic Schr¨ odinger operators. Commun. Math. Phys. 88, 113–131 (1983) 2. P¨ oschel, J: Examples of discrete Schr¨ odinger operators with pure point spectrum. Commun. Math. Phys. 88, 447–463 (1983) 3. Sinai, Ya.G.: Anderson localization for the one-dimensional difference Schr¨ odinger operator with a quasi-periodic potential. J. Stat. Phys. 46 861–909 (1987) 4. Fr¨ ohlich, J., Spencer, T., Wittver, P.: Localization for a class of one-dimensional quasi-periodic Schr¨ odinger operators. Commun. Math. Phys. 132, 5–25 (1990) 5. Eliasson, L.H.: Discrete one-dimensional quasi-periodic Schr¨ odinger operators with pure point spectrum. Acta Math. 179, 153–196 (1997) 6. Goldstein, M.: Anderson localization for quasi-periodic Schr¨ odinger equation. Preprint, 1998. 7. Chulaevsky, V.A., Dinaburg, E.I.: Methods of KAM-Theory for Long-Range Quasi-Periodic Operators on Zν . Pure Point Spectrum. Comm. Math. Phys. 153, 559–577 (1993) 8. Avron,J., Simon, B.: Singular continuous spectrum for a class of almost periodic Jacobi matrices. Bull. Amer. Math. Soc. 6, 81–85 (1982) 9. F. Delyon, D. Petritis: Absence of localization in a class of Schr¨ odinger operators. Commun. Math. Phys. 103, 441–443 (1986) 10. Dinaburg, E.I., Sinai, Ya.G.: The one-dimensional Schr¨ odinger equation with quasi-periodic potential. Funkt. Anal. i. Priloz. 9, 8–21 (1975) 11. R¨ ussmann, H.: On the one-dimensional Schr¨ odinger equation with a quasiperiodic potential.In: Nonlinear Dynamics (Internat. Conf., New York, 1979), Helleman, R.H.G. (ed.) New York: New York Acad. Sci., 1980 12. Moser, J., P¨ oschel, J.: An extension of a result by Dinaburg and Sinai on quasiperiodic potentials. Comment. Math. Helvetici 59, 39–85 (1984) 13. Eliasson, L.H.: Floquet solutions for the one-dimensional quasi-periodic Schr¨ odinger equation. Commun. Math. Phys. 146, 447–482 (1992) 14. Bellissard, J., Lima, R., Testard, D.: A metal insulator transition for the almost Mathieu equation. Commun. Math. Phys. 88, 207–234 (1983) 15. Albanese, C.: KAM theory in momentum space and quasiperiodic Schr¨ odinger operators. Ann. Inst. H. Poincaré, Anal. Non Linéaire 10, 1–97 (1993) 16. Goldstein, M.: Laplace transform methods in the perturbation theory of spectrum of Schr¨ odinger operators I and II. Preprints, 1991–92 17. Krikorian, R.: Réductibilité presque partout des systèmes quasi périodiques analytiques dans le cas SO(3). C. R. Acad. Sci. Paris 321, Série I, 1039–1044 (1995)

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18. Krikorian, R.: Réductibilité des systèmes produits croisés quasi-périodiques ` a ´ valeurs dans des groupes compacts. Paris: Thesis Ecole Polytechnique, 1996 19. Eliasson, L.H.: Ergodic skew-systems on SO(3, R). Preprint ETH-Z¨ urich, 1991 20. Eliasson, L.H.: Reducibility, ergodicity and point spectrum. Proceedings of the International Congress of Mathematicians. Berlin 1998, Vol. II, 779–787 (1998) 21. Young, L.S.: Lyapunov exponents for some quasi-periodic cocycles. Ergod. Th.& Dynam. Syst. 17, 483–504 (1997) 22. Chualevsky, V.A., Sinai, Ya.G.: Anderson localization and KAM-theory. In: P. Rabinowitz, E. Zehnder (eds.): Analysis etcetera. New York: Academic Press, 1989 23. Chualevsky, V.A., Sinai, Ya.G.: The exponential localization and structure of the spectrum of 1D quasi-periodic discrete Schr¨ odinger operators. Rev. Math. Phys. 3, 241–284 (1991) 24. Goldstein, M.: Quasi-periodic Schr¨ odinger equation and analogue of the Cartan’s estimate for real algebraic functions. Preprint, 1998 25. Eliasson, L.H.: Communication at the 4th Quadriennial International Conference on Dynamical Systems, IMPA, July 29–August 8, 1997 26. Bourgain, J.: Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Schr¨ odinger equation. Ann. of Math. 148, 363–439 (1998) 27. Folland, G.D.: Introduction to partial differential equations. Princeton, N.J.: Princeton University Press, 1976 28. Shapiro, H.: Topics in approximation theory, LMN 187. Berlin: Springer-Verlag, 1971 29. MacDonald, I.G.: Symmetric functions and Hall polynomials. Oxford: Clarendon Press, 1979

KAM-persistence of finite-gap solutions Sergei B. Kuksin

Introduction The paper is devoted to a proof of the following KAM theorem: most of space-periodic finite-gap solutions of a Lax-integrable Hamiltonian PDE persist under small Hamiltonian perturbations as time-quasiperiodic solutions of the perturbed equation. In order to prove it we obtain a number of results, important by itself: a normal form for a nonlinear Hamiltonian system in the vicinity of a family of lower-dimensional invariant tori, etc. The paper is an abridged version of the manuscript [KK], missing proofs can be found there. Still our presentation is rather complete modulo a KAM-theorem for perturbations of linear equations (for its proof see [K, K2, P]). Notations. Everywhere in the paper “linear map” means “bonded linear map”. For a linear operator J we denote by J¯ the operator −J −1 . For a complex Hilbert space, ·, · stands for a complex-bilinear quadratic form such that u, u ¯ = u2 . We systematically use the following non-standart notations: for a vector V = (V1 , . . . , Vn ) ∈ Nn we denote NV = {m ∈ N | m = Vj ∀ j} and ZV = {m ∈ Z | m = ±Vj ∀ j}; we abbreviate N(1,2,...,n) to Nn , etc. In particular, Z0 stands for the set of non-zero integers.

1 1.1

Some analysis in Hilbert spaces and scales Smooth and analytic maps

Below we shall work with differentiable maps between domains in Hilbert (more general, Banach) spaces. Since the category of C r -smooth Fréchet maps with r ≥ 2 is rather cumbersome and since only analytic object arise in our main theorems, we mostly restrict ourselves to the two extreme cases: with few exceptions the maps will be either C 1 -smooth or analytic. Now we fix corresponding notations and briefly recall some properties of C 1 -smooth and analytic maps. Let X, Y be Hilbert spaces and O be a domain in X. A continuous map f : O → Y is called continuously differentiable, or C 1 -smooth (in the sense of Fréchet) if there exists a bounded linear map f∗ (x) : X → Y which continuously depends on x ∈ O, such that f (x + x1 ) − f (x) = f∗ (x)x1 + o(x1 X ) provided that x, x + x1 ∈ O. We call f∗ (x) a derivative of f or its tangent map. By f ∗ (x) we denote the adjoint map f ∗ (x) = (f∗ (x))∗ : Y → X. For a real Hilbert space X we denote by X c its complexification , X c = X ⊗R C .


62

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Examples. If X is an L2 -space or a Sobolev space of real-valued functions, then X c is a corresponding space of complex functions. If X is an abstract separable Hilbert space and {φj } is its Hilbert basis, then X = { xj φj | xj ’s are real and |xj |2 < ∞}, while X c = { zj φj | zj ’s are complex and . . . }. Let X c , Y c be complex Hilbert spaces and Oc be a domain in X c . A map f : Oc → Y c is called Fréchet-analytic if it is C 1 -smooth in the sense of real analysis (when we treat X c , Y c as real spaces) and the tangent maps f∗ (x) are complex-linear. Locally near any point in Oc such a map can be represented as a normally convergent series of homogeneous maps (see [PT]). For real Hilbert spaces X, Y and a domain O ⊂ X, a map F : O → Y is analytic if it can be extended to a complex-analytic map F : Oc → Y c , where Oc is a complex neighbourhood of O in X c . A map F : X ⊃ O → Y is called δ-analytic (δ is a positive real number) if it extends to a bounded analytic map (O + δ) → Y c (O + δ is the δneighbourhood of O in X c ). We note that compositions of analytic maps are analytic, as well as their linear combinations. Besides, any analytic map is C k -smooth for every k. There is an important criterion of analyticity: a map f : X c ⊃ Oc → Y c is analytic if and only if it is locally bounded 1 and weakly analytic, i.e., for any y ∈ Y c and any affine complex plane Λ ⊂ X c the complex function Λ ∩ Oc → C, λ → F (λ), yY is analytic in the sense of one complex variable. Even more, it is sufficient to check analyticity of these functions for a countable system y = y1 , y2 , . . . of vectors in Y such that the linear envelope of this system is dense in Y (see [PT]). The Cauchy estimate states that if a map F : X c ⊃ Oc → Y c admits a bounded analytic extension to Oc + δ, then for any u ∈ Oc one has: F∗ (u)X,Y ≤ δ −1

sup

u ∈O c +δ

F (u )Y .

(The estimate readily follows from its one-dimensional version applied to the holomorphic functions Oδ (C) λ → F (u + λx), yY , where xX = yY = 1). In particular, this estimate applies to δ-analytic maps between subsets of real Hilbert spaces. If F : X c ⊃ Oc → Y c is an analytic map and for some point x ∈ Oc the tangent map F∗ (x) is an isomorphism, then by the inverse function theorem in a sufficiently small neighbourhood of x the map F can be analytically inverted. The same is true for real analytic maps. See [PT]. For Banach spaces everything is much the same with one extra difficulty: there is no canonical way to define a complexification X c of a real Banach space X . This difficulty should not worry us since all Banach spaces used 1

that is, any point x ∈ Oc has a neighbourhood, where f is uniformly bounded. In particular, any continuous map is locally bounded.

KAM-persistence of finite-gap solutions

63

below are natural and one can immediately guess the right complexification. For example, if X is the space of bounded linear operators Y1 → Y2 where Y1 , Y2 are Hilbert spaces, then X c is the complex space of linear over reals operators Y1 → Y2c , etc. 1.2

Scales of Hilbert spaces and interpolation

Below we shall study one-dimensional in space partial differential equations in appropriate scales of Sobolev spaces {Xs }, where s ∈ Z or s ∈ R. That is, (i) each Xs is a Hilbert subspace of a Sobolev space of order s, formed by scalar- or vector-functions on a segment [0, T ] (so X0 ⊂ L2 ); (ii) there exist: (1) a Hilbert basis {φk | k ∈ Z0 }, Z0 = Z \ {0}, of the space X0 , (2) an even positive sequence {ϑj | j ∈ Z0 } of linear growth, i.e. ϑj = ϑ−j and C −1 |k| ≤ ϑk ≤ C|k| for all k, such that for any real s the set of vectors {φk ϑ−s k | k ∈ Z0 } form a Hilbert basis of the space Xs . The second assumption implies that for any −∞ < a < b < ∞ the space Xc , c = (1 − θ)a + θb, interpolates the spaces Xa and Xb : in notations of [LM], Xc = [Xa , Xb ]θ . In particular, for any u ∈ Xb holds the interpolation θ inequality: uc ≤ u1−θ a ub . The basis {φk } is called a basis of the scale. The norm and the scalar product in Xs will be denoted · s and ·, ·s = ·, ·Xs ; we abbreviate ·, ·0 = ·, ·. Due to ii), u, u2s = u2s = |uk |2 ϑ2s if u = uk φk . k The scalar product ·, · extends to a bilinear pairing Xs × X−s → R. For any space Xs (real or complex) we identify its adjoint (Xs )∗ with the space X−s . We denote by X−∞ , X∞ the linear spaces X−∞ = ∪Xs , X∞ = ∩Xs . The space X∞ is dense in each Xs . It is formed by smooth (vector-) functions. We shall often treat scales {Xs } as abstract Hilbert scales and do not discuss their embeddings to a scale of Sobolev functions. Example 1.1. Let Xs = H0s (S 1 , R) be the Sobolev space of 2π-periodic functions with zero mean-value. This scale satisfies (i)–(iii). For a basis {φk | k ∈ Z0 } we take the trigonometric basis: 1 1 ϕk = √ cos kx, ϕ−k = − √ sin kx for k = 1, 2, . . . π π

(1.1)

(the minus-sign is introduced for further purposes). For a sequence ϑk we take ϑk = |k|. This choise corresponds to the homogeneous scalar product in Xs , u, vs = (u(s) v (s) )dx. Complexification Xsc of this space is the space H0s (S 1 ; C) of complex Sobolev functions.

64

S. B. Kuksin

Given two scales {Xs }, {Ys } as above and a linear map L : X∞ → Y−∞ , we denote by Ls1 ,s2 ≤ ∞ its norm as a map Xs1 → Ys2 . We say that the map L defines a morphism of order d of the scales {Xs } and {Ys } for s ∈ [s0 , s1 ], if Ls,s−d < ∞ for each s ∈ [s0 , s1 ] with some fixed −∞ ≤ s0 ≤ s1 ≤ +∞. 2 If in addition the inverse map L−1 exists and defines a morphism of order −d of the scales {Ys }, {Xs } for s ∈ [s0 + d, s1 + d], we say that L defines an isomorphism of order d of the two scales. If {Ys } = {Xs }, then an isomorphism L is called an automorphism. We shall drop the specification “for s ∈ [s0 , s1 ]” if the segment [s0 , s1 ] is fixed for a moment or can be easily recovered. If L : Xs → Ys−d is a morphism of order d for s ∈ [s0 , s1 ], then the adjoint maps L∗ : (Ys−d )∗ = Y−s+d → (Xs )∗ = X−s form a morphism of the scales {Ys } and {Xs } of the same order d for s ∈ [−s1 + d, −s0 + d]. We call it the adjoint morphism. A morphism L of a Hilbert scale {Xs }, complex or real, is called symmetric (anti symmetric) if L = L∗ (respectively L = −L∗ ) on the space X∞ . In particular, a linear operator L : Xs0 → Ys0 −d is called symmetric (anti symmetric) if L = L∗ (L = −L∗ ) on the space X∞ . If L is a symmetric morphism of {Xs } of order d for s ∈ [s0 , d − s0 ], then L∗ also is a morphism of order d for s ∈ [s0 , d − s0 ] and L = L∗ as the scale’s morphisms. We call such a morphism selfadjoint. Anti selfadjoint morphisms are defined similar. Example. The operator − defines a selfadjoint automorphism of order 2 of the Sobolev scale {H0s }. The operator ∂/∂x defines an anti selfadjoint automorphism of order one. Linear maps from one Hilbert scale to another obey the Interpolation Theorem: Theorem 1 (see [LM, RS2). ] Let {Xs }, {Ys } be two real Hilbert scales and L : X∞ → Y−∞ be a linear map such that La1 ,b1 = C1 , La2 ,b2 = C2 . Then for any θ ∈ [0, 1] we have La,b ≤ Cθ , where a = aθ = θa1 + (1 − θ)a2 , b = bθ = θb1 + (1 − θ)b2 and Cθ = C1θ C21−θ . This result with Cθ replace by 4Cθ remain true for complex Hilbert scales. In particular, if under the theorem’s assumptions a1 − b1 = a2 − b2 =: d, then L defines a morphism of order d of the scales {Xs }, {Ys } for s ∈ [a1 , a2 ]. Corollary. Let L be a selfadjoint or an anti selfadjoint morphism of a scale {Xs } such that La,b = C < ∞ for some a, b. Then Lθ(a+b)−b,θ(a+b)−a ≤ C for any 0 ≤ θ ≤ 1. Proof. Since (Xa )∗ = X−a and (Xb )∗ = X−b , then C = La,b = L∗ −b,−a = L−b,−a . Now the assertion follows from the Interpolation Theorem. 2

if s0 = −∞, then s > s0 since X−∞ and Y−∞ are given no norms. Similar s < ∞ if s1 = ∞.


65

In particular, an operator L as above defines a morphism of order a − b of the scale {Xs } for s ∈ [−b, a] (or ∈ [a, −b] if −b > a). Let −∞ < a ≤ b ≤ ∞ and Os ⊂ Xs , s ∈ [a, b], be a system of compatible domains (i.e., Os1 ∩ Os2 = Os2 if s1 ≤ s2 ). Let F : Oa → Ya−d be an analytic (or C 1 -smooth) map such that its restriction to the domains Os with a ≤ s ≤ b define analytic (or C 1 -smooth) maps F : Os → Ys−d . 3 Then we say that F is an analytic (or C 1 -smooth) map of order d for a ≤ s ≤ b. Example 1.1, continuation. Let us denote by Π the projector Π : H s → H0s , which sends a function u(x) to u(x) − u(x) dx/2π. The Sobolev spaces H s with s > 1/2 are Banach algebras: uvs ≤ Cs us vs , see [Ad]. Therefore for any segment [a, b], 1/2 < a ≤ b ≤ ∞, the map u(x) → Π◦F (u(x)) where F is a polynomial, defines an analytic map H0s → H0s of order zero for s ∈ [a, b]. If g(x) is any fixed function, then the map u(x) → Π(F (u(x)) + g(x)) is analytic of order zero for s ∈ [a, b] if and only if g ∈ H b . The same is true for a map defined by an analytic function F . More general, this is true for the map u(x) → Π(F (u(x), x)) where F (u, x) is a C b -smooth function of u and x, which is δ-analytic in u with some x-independent δ > 0. 1 Given a C -smooth (or analytic) functional H : Xa ⊃ Oa → R , we identify the adjoint space (Xa )∗ with X−a and consider the gradient map ∇H : Oa → X−a ,

∇H(u), v = H∗ (u)v

∀v ∈ Xa .

If the domain Oa belongs to a system of compatible domains Os (a ≤ s ≤ b) and the gradient map ∇H is a C 1 -smooth (or analytic) map of order dH in this system of domains, we write ord ∇H = dH . In this case the linearised gradient map ∇H(u)∗ : Xs → Xs−dH is bounded for any u ∈ Os with a ≤ s ≤ b. Since ∇H(u)∗ v1 , v2 = d2 H(u)(v1 , v2 ) = ∇H(u)∗ v2 , v1 , then the map ∇H(u)∗ is symmetric and due to the Corollary from Theorem 1 it defines bounded linear maps ∇H(u)∗ : Xs → Xs−dH for s ∈ [dH − d, d] (assuming that 2d ≥ dH ). 1.3

Differential forms

For d ≥ 0 and a domain O in a Hilbert space Xd from a Hilbert scale {Xs } we identify tangent spaces Tx O with Xd and treat differential k-forms on O as continuous functions O × (Xd × · · · × Xd ) −→ R , which are polylinear k

and skew-symmetric in the last k arguments (see more in [Ca, La]). We write def

1-forms as a(x) dx, where a : O → X−d and a(x) dx[ξ] →= a(x), ξ for ξ ∈ Xd . Besides, we write 2-forms as A(x) dx ∧ dx, where def

A(x) dx ∧ dx[ξ, η] →= A(x)ξ, η0 3

for ξ, η ∈ Xd ,

if b = ∞, then a ≤ s < ∞. This agreement applies everywhere below.

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and A(x) : Xd → X−d is a bounded anti selfadjoint operator, i.e., its adjoint A(x)∗ : Xd → X−d equals −A(x). All the forms we consider below are analytic, where a k-form ωk on O ⊂ Xd is called analytic if the corresponding map from O to the linear space of skewsymmetric polylinear functions (Xd × · · · × Xd ) −→ R is analytic. k

To define the differential of a C 1 -smooth k-form ωk we use the Cartan formula: d ωk (x)[ξ1 , . . . , ξk+1 ] =

k+1

(−1)i−1

i=1

∂ ωk (x)[ξ1 , . . . , ξî , . . . , ξk+1 ] . ∂ξi

(1.2)

Here the vectors ξj ∈ Tx O " Xd are extended to constant vector fields on O. So the r.h.s. of (1.2) is well-defined and the commutator-terms from the r.h.s. of the classical Cartan formula (see e.g. [Go, La]) vanish. This definition well agree with the finite-dimensional situation, as states the following obvious lemma: Lemma 1. Let ωk be a k-form on a domain O ⊂ Xd , L be a finite-dimensional affine subspace of Xd and LO = L ∩ O. Then dωk |LO = d(ω |LO ). Proof. Both forms are given by the same formula (1.2).

Example 1.2. 1) The differential of a C 1 -function f on O (=a zero-form) equals df = ∇f (x) dx. 2) The differential of a 1-form a(x) dx, a : O → X−d , equals d(a(x) dx) = (a(x)∗ − a(x)∗ ) dx ∧ dx. Indeed, A(x) = a(x)∗ − the operator a(x)∗ : Xd → X−d is anti selfadjoint and d a(x)dx [ξ, η] = a(x)∗ ξ, η − a(x)∗ η, ξ = A(x)ξ, η. In the sequel we shall also work with k-forms in sub-domains of the direct products Zd , Zd = X × Yd , Zd z = (x, y), where X is a finite-dimensional Euclidean space and Yd is a space from a Hilbert scale {Ys }.4 We write linear operators A in Zd in the block-form, AXX AXY , A= AY X AY Y where AXY : Yd → X, AY X : X → Yd and AXX : X → X, AY Y : Yd → Yd are bounded linear operators. The operator A is anti selfadjoint (with respect to the scalar product in X × Y0 ) if AXY = −AY∗ X and AXX , AY Y are anti selfadjoint operators. Accordingly we write the 2-form A dz ∧ dz as A(z) dz ∧ dz = AXX (x, y) dx ∧ dx + AXY (x, y) dy ∧ dx + +AY X (x, y) dx ∧ dy + AY Y (x, y) dy ∧ dy. 4

Obviously, the spaces {Zs } also form a Hilbert scale.


67

We note that in our notations AY X (x, y) dx∧dy[(δx1 , δy1 ), (δx1 , δy2 )] = AY X δx1 , δy2 Y = −δx1 , AXY δy2 . For sub-domains of the manifolds Yd , Yd = Rn × Tn × Yd = {(p, q, y)},

(1.3)

we use natural versions of the notations given above. The Poincarè lemma states that “locally” each closed form is exact. The proof is constructive and is well applicable to infinite-dimensional problems (see [Ca, La]). We shall need a version of the lemma for a closed 2-form defined in a neighbourhood O ⊂ Yd of the set P × Tn × {0}, where P is a sub-domain of Rn , such that fibres of the natural fibration O → Rn × Tn are convex. Below we state the result, denoting by w points from Rn × Tn : Lemma 2. If ω2 (w, y) is a closed 2-form in O and ω2 (w, 0) = 0, then ω2 = dω1 , where

1

ω1 (w, y)(δw, δy) =

ω(w, ty)[(0, y), (δw, tδy)] dt. 0

In particular, if ω2 = AW W (w, y)dw∧dw+AW Y (w, y) dy∧dw−A!∗W Y (w, y) dw 1 ∧ dy, then ω1 = a(w, y) dw, where a(w, y) = 0 AW Y (w, ty) dt y. This result follows from its finite-dimensional version (see [AG]) and Lemma 1. Indeed, For any (w, y) ∈ O and ξ1 , ξ2 ∈ T(w,y) Yd " R2n × Yd = Zd we denote by Q a sufficiently small neighbourhood of (w, y) in Yd and treat Q as a domain in Zd . Now we take for L the affine 3-space through (w, y) in the directions (0, y), ξ1 , ξ2 and get that dω1 (w, y)[ξ1 , ξ2 ] = ω2 (w, y)[ξ1 , ξ2 ].

2 2.1

Symplectic structures and Hamiltonian equations Basic definitions

¯ dx ∧ In a domain Od ∈ Xd with d ≥ 0 let us take a closed 2-form α2 = J(x) ¯ dx such that the operator J analytically depends on x ∈ Od and defines a ¯ : Xd −→ ∼ Xd+dJ , dJ ≥ 0. Since the operator is anti linear isomorphism J(x) selfadjoint in X0 , then by the Corollary from Theorem 1 it defines an anti selfadjoint automorphism of order −dJ of the scale {Xs } for s ∈ [−d − dJ , d]. The form α2 supplies Od with a symplectic structure which corresponds to any C 1 -smooth function h on Od the Hamiltonian vector field Vh defined by the usual (see [A1]) relation: α2 [Vh , ξ] = −dh(ξ) for all ξ ∈ T Od . For any ¯ x ∈ Od we have J(x)V h (x), ξ = −∇h(x), ξ for each ξ ∈ Xd . Thus, Vh (x) = J(x)∇h(x),

where

¯ −1 . J = (−J)

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S. B. Kuksin

The operator J(x) is called an operator of the Poisson structure. For each x it defines an anti selfadjoint automorphism of the scale of order dJ and the operator −→

J(x) : Xs+dJ ∼ Xs ,

−d − dJ ≤ s ≤ d,

(2.1)

analytically depend on x ∈ Od (see the Corollary to Theorem 1). Since the functional h is C 1 -smooth, then the gradient map ∇h : Od → X−d is continuous. Using (2.1) we get that the vector field Vh defines a continuous map Od → X−d−dJ . Usually we shall impose an additional restriction and assume that the vector field Vh is smoother than that and ord Vh =: d1 < 2d + dJ . To stress that a domain Od ⊂ Xd is given a symplectic structure as above we shall write it as a pair (Od , α2 ). If the form α2 is analytic on the whole space Xs for each s ≥ s0 with some fixed s0 , we shall say that ({Xs }, α2 ) is a symplectic Hilbert scale. A basis {φj (x) | j ∈ Zn } of a tangent space Tx Od = Xd is called symplectic if α2 [φj (x), φk (x)] = νj (x)δj,−k

(2.2)

for any j ∈ N and each k ∈ Z0 , with some positive real numbers νj (x), j ∈ N. For any C 1 -smooth function h on Od × R a Hamiltonian equation with the hamiltonian h(x, t) is the equation x˙ (t) = J(x)∇h(x, t) =: Vh (x, t).

(2.3)

If ord Vh = 0 and the vector field Vh is C 1 -smooth and Lipschitz in Od , then the initial-value problem for the equation (2.3) is well-posed: for any given initial condition x(0) ∈ Od it has a unique solution defined while it stays in Od . A partial differential equation, supplemented by appropriate boundary conditions, is called a Hamiltonian PDE if under a suitable choice of a symplectic Hilbert scale ({Xs }, α2 ), a domain Od ⊂ Xd and a hamiltonian h, the equation can be written in the form (2.3). In this case the vector field Vh is unbounded, ord Vh = d1 > 0: Vh : Od × R → Xd−d1 .

(2.4)

Usually the domain Od belongs to a system of compatible domains Os , s ≥ d0 , and the map Vh is analytic of order d1 for s ≥ d0 . For a vector field Vh as in (2.4) with d1 > 0, different classes of solutions for (2.3) can be considered. We choose the following definition: a continuous curve x : [0, T ] → Od is a solution of (2.3) in a space Xd if it defines a C 1 -smooth map [0, T ] → Xd−d1 and both parts of (2.3) coincide as curves in Xd−d1 . A solution x(t) is called smooth if it defines a smooth curve in each space Xl . If a solution x(t), t ≥ τ , of (2.3) with x(τ ) = xτ exists and is unique, we write x(t) = Sτt xτ , or x(t) = S t−τ xτ if the equation is autonomous. The operators Sτt and S τ are called flow-maps of the equation.


69

For an equation (2.3) with d1 > 0 there is no general existence theorem for a solution of the corresponding initial-value problem which would guarantee existence of the flow-maps. To prove the existence is an art we do not touch here. Example 2.1 (semilinear equation). If (2.3) is a semilinear equation, i.e., Vh = A + V 0 , where A is an unbounded linear operator with a discrete imaginary spectrum and the nonlinearity V 0 is Lipschitz on bounded subsets of Xd , then solutions of the equation with prescribed initial conditions are well defined till they stay in a bounded part of Xd , see [Paz]. Some important Hamiltonian PDEs are semilinear. For example, the nonlinear Schr¨ odinger equation: u(t, ˙ x) = i(∆u + f (|u|2 )u), x ∈ S 1 , where f is a smooth real-valued function (see [K, Paz]). Still, the semilinearity assumption is very restrictive since it fails for many important Hamiltonian PDEs (e.g., for the KdV). Our main concern are quasilinear Hamiltonian equations, i.e., equations (2.3) with Vh = A + V 0 , where A is a liner operator and ord A >ord V 0 . Possibly ord V 0 > 0 i.e., the equation may be non-semilinear. Let Q ⊂ Od be a sub-domain such that the flow-maps maps Sτt : Q → Od are well-defined and are C 1 -smooth for τ ≤ t ≤ T , where −∞ ≤ τ < T ≤ ∞. Then differentiating a solution x(t) of (2.3) in the initial condition we get that the curve ζ(t) := Sτt x(τ ) ∗ ζ satisfies the linearised equation ˙ = Vh (x(t), t)∗ ζ(t), ζ(t)

ζ(τ ) = ζ.

(2.5)

The assumption that the map Sτt is C 1 -smooth in a sub-domain is very restrictive since to check the smoothness of flow-maps for many important equations (even for the KdV!) is a nontrivial task. To get rid of it we give the following Definition 1. Let x(t), t ∈ R, be a solution for equation (2.3). If for each ζ ∈ ˙ = Vh (x(t), t)∗ ζ(t), ζ(θ) = ζ, has Xd and each θ the linearised equation ζ(t) a unique solution ζ(t) ∈ Xd defined for all t and such that ζ(t)d ≤ Cζd t (x)ζ and uniformly in θ, t from a compact segment, then we write ζ(t) = Sθ∗ t say that flow {Sθ∗ (x)} of the linearised equation (2.5) is well defined in Xd . The property described in Definition 1 characterises the flow only in the “infinitesimal vicinity” of a solution of (2.3). It suits well our goal to study special families of solutions rather than the whole flow of the equation. If the flow-maps Sτt are C 1 -smooth, then Sτt (x)∗ = Sτt ∗ (x), but the map in the r.h.s. of this relation can be well defined while the map in the l.h.s. is not. Example 2.2 (Equations of the Korteweg–de Vries type). Let us take for {Xs } the scale of Sobolev spaces H0s as in Example 1.1. We define a ∂ Poisson structure by means of the operator J = ∂x , so dJ = 1 and −J¯

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S. B. Kuksin

is the operator (∂/∂ x)−1 of integrating with zero mean-value. We get the symplectic Hilbert scale ({H0s }, −(∂/∂ x)−1 du ∧ du). We stick to the discrete scale {s ∈ Z}: it is sufficient since the orders of all involved operators are integer. The trigonometric basis {ϕj | j ∈ Z0 } (see (1.1)) is symplectic since for j ≥ 1 and any k we have: ¯ −1/2 cos jx, ϕk (x) = j −1 −π −1/2 sin jx, ϕk (x) = j −1 δj,−k . α2 [ϕj , ϕk ] = Jπ 2π For a hamiltonian h we take h(u) = 0 (− 18 u (x)2 + f (u)) dx with some analytic function f (u). Then ∇h(u) = 14 u + f (u) 5 and Vh (u) = 14 u + ∂ ∂x f (u). Thus the Hamiltonian equation takes the form u(t, ˙ x) =

1 ∂ u + f (u) 4 ∂x

(2.6)

1 3 The maps H0s → R, u(x) → (for f (u) = 4 u swe get sthe KdV equation). f (u) dx and H0 → H , u(x) → f (u(x)), with s ≥ 1 are analytic (see Example 1.1). So the map H0s u → Vh (u) ∈ H0s−3 is analytic for s ≥ 1. That is, the vector field Vh is analytic of order 3 for s ≥ 3. Being supplemented by an initial condition u(0, x) = u0 (x) ∈ H0s with s ≥ 3, equation (2.6) has a unique solution in H0s . This solution exists for t < T (u0 s ) (T is a continuous positive function) and the flow-maps S t , t < T , are C 1 -smooth. This is a non-trivial result, see e.g. [Kat1]. On the contrary, if u(t, x) is a smooth solution of (2.6), then the linearised equation

v˙ =

∂ 1 v + (f (u)v), 4 ∂x

v(0, x) = v0 ∈ H0s ,

(2.7)

has a unique solution in H0s with any s ≥ 0 by trivial arguments, see [KK, Paz]. We shall often work with equations in a sub-domain Od of the manifold Yd (d ≥ 0) as in (1.3), given a symplectic structure by means of a 2-form (dp ∧ ¯ dq) ⊕ (Υ(y)dy ∧ dy), where dp ∧ dq is the classical symplectic form on Rn × T n ¯ and Υ(y)dy∧dy is a closed 2-form in a domain in Yd . This symplectic structure corresponds to a C 1 -smooth function H(p, q, y) the following Hamiltonian system: p˙ = −∇q H, q˙ = ∇p H, y˙ = Υ∇y H. Solutions of these equations are defined in the same way as solutions of (2.3). 2.2

Symplectic transformations

Let {Xs }, {Ys } be two Hilbert scales and d, d˜ ≥ 0. Let O ⊂ Xd and Q ⊂ Yd˜ ¯ be two domains given symplectic structures by 2-forms α2 = J(x)dx ∧ dx and 5

since dh(u)v =

− 14 u (x)v (x) + f (u(x))v(x) dx = 14 u (x) + f (u(x)), v(x)L2 .


71

¯ β2 = Υ(y)dy ∧ dy as in Section 2.1. A C 1 -smooth map Φ : Q → O is called symplectic if Φ∗ α2 = β2 . That is, if for any y ∈ Q with Φ(y) = x ∈ O we have ¯ ¯ J(x)Φ ∗ (y)ξ, Φ∗ (y)ηX0 = Υ(y)ξ, ηY0 for all ξ, η ∈ Yd˜, or ¯ ◦ Φ∗ (y) = Υ(y). ¯ Φ∗ (y) ◦ J(x)

(2.8)

A symplectic map Φ is an immersion since by (2.8) its tangent maps are embeddings. If a symplectic map Φ is such that the tangent maps Φ∗ (y) define isomorphisms of the spaces Yd˜ and Xd , then Φ is called a symplectomorphism. We shall need an obvious version of the definitions above for the case when Oc and Qc are complex domains in Xdc and Yd˜c respectively and the forms α2 , β2 have constant coefficients: an analytic map Φ1 : (Qc , α2 ) → (Oc , β2 ) ¯ 1 (y)∗ ξ, Φ1 (y)∗ ηX ≡ Υξ, ¯ ηY . In particular, an analytic is symplectic if JΦ 0 0 symplectomorphism Φ : Q → O analytically extends to a symplectomorphism of sufficiently small complex neighbourhoods of Q and O (the forms α2 , β2 are assumed to be constant-coefficient). From now on for the sake of simplicity we restrict ourselves to the case we need below: d = d˜ ≥ 0,

¯ = ord Υ(y) ¯ ord J(x) = −dJ

∀ x, y.

¯ = J¯ and Υ(y) ¯ ¯ are constants Proposition 1. Let us assume that J(x) =Υ isomorphisms of the corresponding scales of order −dJ . Let Φ : (Qc , β2 ) → (Oc , α2 ) be a symplectomorphism such that Φ(y)∗ d,d , (Φ(y)∗ )−1 d,d ≤ C for every y ∈ Qc . Then Φ(y)∗ θ,θ , (Φ(y)∗ )−1 θ,θ ≤ C1 for every y ∈ Qc and every θ ∈ [−d − dJ , d]. If Φ is an analytic symplectomorphism, then these maps are analytic in y ∈ Qc . ¯ ∗ ◦J. So Φ(y)∗ d+d ,d+d ≤ C for every Proof. By (2.8) we have Φ∗ = −Υ◦Φ J J y. Hence, Φ(y)∗ −d−dJ ,−d−dJ ≤ C and the estimate for Φ∗ θ,θ follows by interpolation. The estimates !for Φ−1 follow from the identity (Φ∗ )−1 = ∗ ∗ −J¯◦Φ∗ ◦Υ since Φ−1 = (Φ∗ )−1 = (−J¯◦Φ∗ ◦Υ)∗ is a zero-order morphism ∗

for s ∈ [−d − dJ , d]. In the analytic case the maps are analytic in y by the criterion of analyticity. As in the finite-dimensional case, symplectic maps transform Hamiltonian equations to Hamiltonian. Let Φ : Q → Oc be an analytic symplectic map such that Φ(y)∗ : Ys → Xs is a linear map, analytic in y ∈ Q, for any |s| ≤ d. (2.9) ¯ ¯ ¯ are constant isomorphisms of zero We note that if J(x) = J¯ and Υ(y) =Υ order, then the assumption (2.9) is satisfied due to Proposition 1. Theorem 2. Let domains O ⊂ Xd and Q ⊂ Yd , d ≥ 0, be given symplectic ¯ = −dJ . structures by analytic 2-forms α2 , β2 as above with ord J¯ =ord Υ

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Let the vector field Vh = J∇h of equation (2.3) defines a C 1 -smooth map Vh : O × R → Xd−d1 of order d1 ≤ 2d and let Φ : Q → O be an analytic symplectic map satisfying (2.9), such that the vector field Vh in O is tangent to Φ(Q) in the following sense: Vh (Φ(y)) = Φ(y)∗ ξ

for any y ∈ Q with an appropriate ξ ∈ Yd−d1 . (2.10)

y˙ = Υ(y)∇y H(y, t),

¯ −1 , H = h ◦ Φ , Υ = (−Υ)

(2.11)

to solutions of (2.3). We note right away that the assumption (2.10) becomes empty if Φ is a symplectomorphism (to be more specific, now (2.10) follows from (2.9) since d − d1 ≥ −d). Proof. Let y(t) be a solution of (2.11). By (2.9) the curve x(t) = Φ(y(t)) is C 1 -smooth in Yd−d1 and is continuous in Yd . It remains to check that it satisfies (2.3). Since x˙ = Φ(y)∗ y˙ and ∇y H = Φ(y)∗ ∇x h, then ¯ x˙ = Φ(y)∗ Υ(y)Φ(y)∗ ∇x h = −Φ(y)∗ Υ(y)Φ(y)∗ J(x)V h (x). ¯ By (2.10), Vh (x) = Φ(y)∗ ξ. So the r.h.s is −Φ(y)∗ Υ(y)Φ(y)∗ J(x)Φ(y) ∗ ξ. By ¯ (2.8) it equals −Φ(y)∗ Υ(x)Υ(x)ξ = Φ(y)∗ ξ = Vh (x). Thus, x(t) satisfies the equation (2.3). To apply Theorem 2 we have to be able to construct sufficient amount of symplectic transformations. An important way to construct symplectomorphisms of sub-domains of (O ⊂ Xd , α2 ) is to get them as flow-maps Stτ of an additional nonautonomous Hamiltonian equation x˙ = J(x)∇x f (t, x) = Vf (t, x), where the hamiltonian f is such that the vector field Vf is Lipschitz, see [K, KK]. ¯ dx ∧ dx). Let O be a domain in a symplectic space (Xd , α2 = J(x) Definition 2. Let C 1 -smooth functions H1 , H2 on a domain O ⊂ Xd define continuous gradient maps of orders d1 , d2 ≤ 2d such that d1 + d2 + dJ ≤ 2d. Then the Poisson bracket {H1 , H2 } of H1 , H2 is the continuous on O function {H1 , H2 }(x) = J(x)∇H1 (x), ∇H2 (x). The scalar product J(x)∇H1 , ∇H2 is well-defined and is continuous in x. The Poisson bracket is skew-symmetric, {H1 , H2 } = −{H2 , H1 }. In particular, {H, H} ≡ 0 (if ord ∇H ≤ d − dJ /2). 2.3

Darboux lemmas

The classical Darboux lemma states that locally near a point any closed nondegenerate 2-form in R2n can be written as dp ∧ dq. This result has several


73

versions which put a closed non-degenerate 2-form on a manifold to different normal forms in the vicinity of a closed set (for the classical lemma the set is a point), see [AG]. Some of these results admit direct infinite-dimensional reformulations which can be proven by the same arguments due to Moser– Weinstein. In this section we present a version of the Darboux lemma which will be used later on. Let Yd = Rn × T n × Yd and W be its subset of the form W = P × n T × {0}, where P is a bounded sub-domain of Rn . By O, O1 , . . . we denote δ-neighbourhoods of W in Yd with different δ > 0 and suppose that in a neighbourhood O we are given two closed analytic 2-forms ω0 and ω1 , both of them of the form ωj = J¯j (y) dy ∧ dy, where y = (p, q, y) and dy = (dp, dq, dy). We assume that (i) ω0 = ω1 in T O |W , (ii) for all t ∈ [0, 1] and all y ∈ O the map J¯t (y) = (1 − t)J¯0 + tJ¯1 defines an isomorphism J¯t : Zd →−→ Zd+dJ with some fixed dJ ≥ 0, where ∼ Zd = Rn × Rn × Yd . By (ii), the map J t = (−J¯t )−1 : Zd+d →−→ Zd is well defined and J

∼

analytically depends on y. By Poincaré’s lemma (see Lemma 2 above), the form ω1 − ω0 equals dα for some analytic one-form α = a(y) dy such that a(p, q, y) = O(y2d ). Let us also assume that (iii) the map O1 → Zd+dJ , y → a, is Lipschitz analytic in a neighbourhood O1 of W . Lemma 3 (Moser–Weinstein). Under the assumptions i)-iii) there exists a neighbourhood O2 and an analytic diffeomorphism ϕ : O2 → O such that ϕ |W = id, ϕ∗ |W = id and ϕ∗ ω1 = ω0 . Moreover, ϕ equals to a time-one flow-map S01 , corresponding to the non-autonomous equation y˙ = J t a(y) =: V (t, y). Proof. For 0 ≤ t ≤ 1 let us consider the 2-forms ωt = (1 − t)ω0 + tω1 = J¯t dy ∧ dy. These forms are closed as well as the forms ω0 , ω1 and are nondegenerate in a neighbourhood O3 since ωt = ω0 = ω1 on W by i). Now we denote by ϕt the flow-maps S0t of equation y˙ = V (t, y); so ϕ0 = id, ϕ1 = ϕ and (ϕ1 − id)(p, q, y) = O(y2d ). The lemma will be proven if we check that (ϕt )∗ ωt = const. Because Cartan’s identity (see e.g. [GS] and see [KK] for t the infinite-dimensional case) we have to prove that ∂ω ∂t + d(V %ωt ) = 0. Since ∂ωt /∂t = ω1 − ω0 = dα, then it remains to check that α + V %ωt ≡ 0. But V %ωt = V %J¯t dy ∧ dy = (J¯t V )dy. So α + V %ωt = (a + J¯t V ) dy = (a + J¯t J t a) dy = 0 and the lemma is proven. Appendix. Time-quasiperiodic solutions Our main goal is to study time-quasiperiodic solutions x(t) of some Hamiltonian equations (2.3). Here we recall corresponding basic definitions.

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Definition. A C 1 -curve γ : R → X in a Banach space or a manifold X is called quasiperiodic (QP) with ≤ n frequencies if there exists a C 1 -smooth map Γ : T n → X, a vector ω ∈ Rn and a point q0 ∈ T n such that γ(t) ≡ Γ(q0 + ωt) .

(A.1)

The vector ω is called the frequency vector and q0 is called the phase. The minimal n such that γ(t) admits a representation (A.1) is called the number of independent frequencies; corresponding numbers ω1 , . . . , ωn are called the basic frequencies and the numbers 2π/ω1 , . . . , 2π/ωn – the basic periods. Remark. We note that the vector ω formed by the basic frequencies is defined only up to an unimodular transformation L since the curve γ(t) can be also written as γ(t) = ΓL (Lq0 + Lωt), where ΓL (q) = Γ(L−1 q). What is uniquely defined, is the Z-module Z ω1 + Z ω2 + · · · + Z ωn ⊂ R. We shall usually ignore this subtlety. The closure γ(R) is called the hull of γ. If components of ω are rationally independent, then the hull equals Γ(Tn ). We call a solution x of (2.3) a (time-) quasiperiodic, or a QP solution, if the curve x : R → Xd is QP, and call it analytic quasiperiodic if the corresponding map Γ is analytic of maximal rank. The hull Γ(T n ) of an analytic QP solution (A.1) with n basic frequencies is an invariant analytic n-torus of the equation. This torus is an analytic submanifold of X if the map Γ is an immersion.

3

Lax-integrable Hamiltonian equations and their integrable subsystems

We consider a symplectic Sobolev scale ({Zs }, α2 ), α2 = J¯ dz ∧ dz, as above. The operator J¯ defines an anti selfadjoint automorphism of the scale of a negative order −dJ ≤ 0. To a hamiltonian H = 12 Az, z + H(z), where A is a selfadjoint morphism of the scale of order dA , the symplectic structure corresponds the Hamiltonian equation u˙ = J∇H(u) = J(Au + ∇H(u)) =: VH (u),

¯ −1 . J = (−J)

(3.1)

We assume that the hamiltonian H is analytic quasilinear, that is, the functional H is analytic on a domain Od ⊂ Zd , d ≥ dA /2, and defines an analytic gradient map of order dH < dA , ∇H : Od → Zd−dH . By this assumption and the Corollary to Theorem 1, for any u ∈ Od the linear map ∇H(u)∗ defines a morphism of the scale {Zs } of order dH for s ∈ [−d + dH , d]. We shall study equation (3.1) in a sufficiently smooth space Zd taking any d such that d1 := ord VH = dA + dJ ≤ 2d + dJ . We do not assume that the flow maps of the equation are defined on the whole domain Od (i.e., we do not assume that the equation can be solved for any initial condition u(0) ∈ Od ).


3.1

75

Examples of Hamiltonian PDEs

Quasilinear Hamiltonian PDEs with analytic coefficients have the form (3.1), where usually Od equals to the whole space Zd and the gradient map ∇H is analytic of some order dH for all sufficiently smooth spaces Zd (i.e., it is an analytic map of order dH for d ≥ d0 with some fixed d0 ). The following examples and their perturbations will be the main through our work: Example 3.1 (KdV equation, cf. Example 2.2). Let us take for a scale {Zs | s ∈ Z} the scale {H0s (S 1 ; R)} of 2π-periodic Sobolev functions with zero mean value, defined in Example 1.1. We choose! J = ∂/∂x, A = 14 ∂ 2 /∂x2 and 2 − 18 u + 14 u3 dx. The equation (3.1) takes H(u) = 14 u3 dx, so H(u) = the form: u˙ =

1 ∂ (uxx + 3u2 ). 4 ∂x

(KdV)

It is considered under zero mean-value periodic boundary conditions: u(t, x) ≡ u(t, x + 2π),

2π

u(t, x) dx ≡ 0, 0

which are satisfied automatically since we are looking for solutions in a space H0s . The gradient map Zd → Zd , u → ∇H = 34 u2 , is analytic of order dH = 0 for d ≥ 1 (see in Example 2.2). Now we have ord A = dA = 2 and ord J = dJ = 1. Example 3.2 (higher KdV equations). The KdV equation in the previous example is an equation from an infinite hierarchy of Hamiltonian equations, called the KdV-hierarchy [DMN, McT, ZM]. The l-th equation from the hierarchy can be written as an equation (3.1) in the same symplectic Hilbert scale ({H0s }, J du, du). It has a hamiltonian Hl of the form Hl (u) = 2π ! 2 Kl (−1)l u(l) + higherordertermswith ≤ l − 1 derivatives dx, 0

where Kl is a non-zero constant (H1 is just the KdV-hamiltonian). In particular, the hamiltonian H2 has the form H2 = 18 (u2xx − 5u2 uxx − 5u4 ) dx and the corresponding Hamiltonian equation is the fifth order partial differential equation: 1 ∂ 1 u˙ = u(5) − (5u2x + 5uuxx + 10u3 ). 4 4 ∂x The gradient map of the non-quadratic part of this hamiltonian defines in the Sobolev scale {H0s } an analytic map of order dH = 2 for any s ≥ 2. The order dA of the linear part equals 4 and dJ = 1.

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Example 3.3 (Sine-Gordon equation). The Sine-Gordon (SG) equation on the circle, u ¨ = uxx (t, x) − sin u(t, x),

x ∈ S = R/2πZ,

(SG)

corresponds to a bounded nonlinearity. For any u0 ∈ H s (S) and u1 ∈ H s−1 (S) this equation has a unique solution u(t, x) ∈ C(R, H s ) ∩ C 1 (R, H s−1 ) such that u(0, x) = u0 and u(0, ˙ x) = u1 . This is almost obvious, see [Paz]. The equation (SG) can be written in a Hamiltonian form in many different ways. The most strightforward way is to write (SG) as u˙ = −v,

v˙ = −uxx + sin u(t, x).

(3.2)

To are Hamiltonian, we take the symplectic scale see thats these equations ¯ dξ , where ξ = (u, v) ∈ Zs and {Zs = H (S) × H s (S)}, α2 = Jdξ, J(u, v) = (−v, u) (so J¯ = J). For a hamiltonian H we choose H = 12 Aξ, ξ + H(ξ), where A(u, v) = (−uxx , v) and H(u, v) = − cos u(x) dx. Then ∇H(u, v) = (sin u, 0) and the Hamiltonian equation ξ˙ = J∇H = J(Aξ + ∇H(ξ)) coinsides with (3.2). The Hamiltonian form (3.2) is traditional (cf. [McK]) and is convenient to study explicit (“finite-gap”) solutions of (SG), but not to carry out detailed analysis of the equation since the linear operator A as above defines a selfadjoint morphism of the scale {Zs } of order two, which is not an order-two automorphism (the invererse map A−1 defines a morphism of order 0, not −2). To get a Hamiltonian form of the SG equation, convenient for its analysis, we denote w = A−1/2 v, where A = −∂ 2 /∂x2 + 1, and write (SG) as √ √ u˙ = − A w, w˙ = A u + A−1 (sin u − u) . (3.3) The equations (3.3) have more symmetric form than (3.2). They turn out to be Hamiltonian in the shifted Sobolev scale {Zs = H s+1 (S) × H s+1 (S)}, see [K, BiK1] and [KK]. Even periodic boundary conditions. If u(t, x) is any solution of (SG) such that the initial conditions (u(0, x), u(0, ˙ x)) = (u0 (x), u1 (x)) are even periodic functions, i.e., u0 (x) ≡ u0 (x + 2π) ≡ u0 (−x)

(EP)

and similar with u1 , then u− (t, x) = u(t, −x) is another 2π-periodic solution for SG with the same initial conditions. Since a solution of the initial-value problem for (SG) is unique, then u− (t, x) ≡ u(t, x). That is, the space of even periodic functions is invariant under the SG-flow and we can study the equation under the boundary conditions (EP). These conditions clearly imply Neuman boundary conditions on the half-period: (N) u0x , u1x (0) = u0x , u1x (π) = (0, 0).


77

The former can be viewed as a “smoother version” of the latter since for any smooth even periodic function all its odd-order deviatives (not only the first one) coinside at x = 0 and x = π. Denoting for any real s by Zse a subspace of Zs , formed by even functions, we observe that the equation (SG)+(EP) can be written in the Hamiltonian ¯ dξ , where Z e = {ξ(x) ∈ form (3.2) in the symplectic scale {Zse }, Jdξ, s Zs | ξ satisfies (EP). We note that for s = 2 the space Z2e is formed by the vector-functions from H 2 (0, π) × H 2 (0, π) which satisfy (N) (the functions are assumed to be extended to the segment [0, 2π] in the even way). That is, for solutions of the SG equation in the Sobolev space H 2 , the boundary conditions (OP) and (N) are equivalent. Odd periodic boundary conditions. Similarly, the SG-equation under the odd periodic boundary conditions u(t, x) ≡ u(t, x + 2π) ≡ −u(t, −x)

(OP)

can be written in a Hamiltonian form in the symplectic scale {Zso }, β2 = ¯ dξ , where Z o = {ξ(x) ∈ Zs | ξ satisfies (OP) }. These boundary Jdξ, s conditions imply the Dirichlet ones: u0 , u1 (0) = u0 , u1 (π) = (0, 0). 3.2

Lax-integrable equations

Let us consider a quasilinear Hamiltonian PDE and write it in the Hamiltonian form (3.1). This equation is called Lax-integrable if there exist linear operators Lu , Au which depend on u ∈ Z∞ and define linear morphisms of finite orders of some additional real or complex Hilbert scale {Zs }, such that u(t) is a smooth solution of (3.1) if and only if d Lu(t) = [Au(t) , Lu(t) ]. dt

(3.4)

The operators Lu and Au are said to form an L-A pair of the equation (3.1)=(3.4). We specify dependence of the L, A-operators on u and assume existence of integers s and d such that for all s ≥ s the maps u → Lu and u → Au are analytic as maps from Zs to the space of bounded linear operators Zd → Zd−d , provided that d ≤ s. Due to this assumption, the l.h.s of (3.4) is well defined for any C 1 -smooth curve u(t) ∈ Zs , s ≥ s . Both KdV and Sine-Gordon equations are Lax-integrable, see below Section 4. We abbreviate Lt = Lu(t) and At = Au(t) , where u(t) is a smooth solution for (3.4). A crucial property of the L, A-operators is that spectrum of the operator Lt is time-independent and its eigen-vectors are preserved by the flow, defined by the operators At :

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Lemma 4. Let χ0 ∈ Z∞ be a smooth eigenvector of L0 , L0 χ0 = λχ0 . Let us also assume that the initial-value problem χ˙ = At χ,

χ(0) = χ0 ,

(3.5)

has a unique smooth solution χ(t) ∈ Z∞ . Then Lt χ(t) = λχ(t) for any t. Proof. Let us denote Lt χ(t) = ξ(t), λχ(t) = η(t) and calculate derivatives of these functions. We have: d d ξ = Lχ = [A, L]χ + LAχ = ALχ = Aξ dt dt d d and dt η = dt λχ = λAχ = Aη . Thus, both ξ(t) and η(t) solve the problem (3.5) with χ0 replaces by λχ0 and coincide for all t by the uniqueness assumption.

In many important examples of Lax-integrable equations, {Zs } is the Sobolev scale of L-periodic in x (vector-) functions and L, A are u-dependent differential operators, acting on complex vector-functions. In this case it is natural to take for the scale {Zs } the Sobolev scale of L-periodic complex vector-functions. So L-periodic spectrum of the operator Lu is an integral of motion for the equation (2.8) if the linear equation (2.11) defines a flow in the space of smooth L-periodic vector-functions. The set of integrals which we obtain in this way usually is incomplete. To get missing integrals we note that an L-periodic in x solution u(t, x) can be also treated as an Lm-periodic solution for any m ∈ N. Accordingly, we can consider the L, A-operators under mL-periodic boundary conditions and take for {Zs } the Sobolev scale of mL-periodic vector-functions. Due to the lemma, the mL-periodic spectrum of L is an integral of motion if the equation (2.11) defines a flow in the corresponding space Z∞ . This set of integrals contains the initial one. 3.3

Integrable subsystems

Now we suppose that equation (3.1) possesses an invariant submanifold T 2n ⊂ Od ∩ Z∞ , such that restriction of the equation to T 2n is integrable. For some important examples the manifold T 2n may have singularities and the restricted symplectic form α2 |T 2n may degenerate. Since our objects are analytic, these degenerations can only happen on singular subsets of positive codimension and do not affect the final KAM-results which neglect subsets of small measure: at some point we shall cut out the singular subsets with their small neighbourhoods. But our preliminary arguments are global . To carry them out we have to develop global notations. We assume that T 2n = Φ0 (R × T n ) where T n = {z} is the standard ntorus and R is a connected n-dimensional analytic set which is the real part of a connected complex analytic subset Rc of a domain Πc ⊂ CN . 6 By Rsc


79

we denote any proper analytic subset of Rc which contains its singular part and denote by Rs the real part of Rsc , i.e., Rs = Rsc ∩ R. We assume that the map Φ0 is analytic and the form α2 |T 2n does not degenerate identically: The map Φ0 : R × T n → Zl is analytic for each l. That is, for some δ = δl > 0 it extends to an analytic map Πc × {|Im z| < δ} → Zlc . c such that the analytic 2-form (ii) Rc contains a proper analytic subset Rs1 ∗ c Φ0 α2 is non-degenerate in (R \ (Rs ∪ Rs1 )) × T n , where Rs1 = Rs1 ∩ R. (i)

For brevity we re-denote Rs := Rs ∪ Rs1 and similar re-denote Rsc . We set R0c = Rc \ Rsc and R0 = R \ Rs . Since Rs and Rsc comprise singularities of the analytic sets R and Rc as well as of the map Φ0 , then the sets R0 and R0c are smooth analytic manifolds and the map Φ0 : R0 × T n → Zl , Φ0 (R0 × T n ) =: T02n , is an immersion. We assume that (iii) The set T02n is a smooth analytic submanifold of each space Zl , invariant for the equation (3.1), as well as the tori T n (r) = Φ0 ({r} × T n ), r ∈ R0 . The restricted to T n (r) equation takes the form z˙ = ω(r), where ω extends to an analytic map Πc → Cn . Due to (ii) and (iii), the manifold T02n is filled with smooth time-quasiperiodic solutions of the equation (3.1). The frequency map r → ω(r) is assumed to be non-degenerate: (iv) for almost all r ∈ R0 , the tangent map ω∗ (r) : Tr R0 → Rn is an isomorphism. By Theorem 2 the equation restricted to T02n is Hamiltonian. Condition iv) implies that this equation is integrable (see [KK] or [BoK2], p. 106). So by the Liouville–Arnold theorem, R0 can be covered by a countable system of open domains R0j , R0 = R01 ∪ R02 ∪ · · · , such that the equation (3.1) restricted to each manifold Tj2n = Φ0 (R0j × T n ) admits action-angle variables (p, q) with actions p ∈ Pj Rn and angles q ∈ T n . I.e., ω2 = dp ∧ dq and the equation restricted to Tj2n takes the form p˙ = 0, q˙ = ∇h(p),

h = H ◦ Φ0 ,

where h = hj . Lemma 5. In any domain R0j we have ∇h(p(r)) = ω(r) and q = z + q 0 (r). For many important examples the set of integrals of motion of a Laxintegrable equation, formed by 2π-periodic and 2π-antiperiodic spectra (see Lemma 4 and its discussion), is “complete”. Fixing all but some n of them one can construct invariant manifolds T 2n as above. Below we show how to carry out this construction for KdV and SG equations. 6

That is, Rc is formed by zeroes of an analytic map Πc → CN −n such that at some points of Πc its linearisation has full rank. For elementary facts concerning analytic sets, real and complex, see [Mil] and [GR], Sections II, III.

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S. B. Kuksin

Finite-gap manifolds and theta-formulas

We start with famous finite-gap solutions for the KdV equation under zeromeanvalue periodic boundary conditions: 2π 1 ∂ 2 (uxx + 3u ), u(t, x) ≡ u(t, x + 2π), u dx ≡ 0. (KdV) u˙ = 4 ∂x 0 The finite-gap solutions fill invariant submanifolds T 2n ⊂ H0s (S 1 ) with integrable dynamics on them, as in Section 2. To study the manifolds T 2n we shall use the Its - Matveev formula which represents the finite-gap solutions in terms of theta-functions. 4.1

Finite-gap manifolds for the KdV equation 2

∂ The L, A-operators for the KdV equation are Lu = − ∂x 2 − u and Au = ∂3 3 ∂ 3 + u + u . Indeed, calculating the commutator [A, L]v one sees that ∂x3 2 ∂x 4 x most of the terms cancel and there is nothing left except ( 14 uxxx + 32 uux )v. Thus, [A, L] is an operator of multiplication by the r.h.s. of KdV and the equation can be written in the form (3.4). For the scale {Zs } we take one of the following scales of complex functions: or the scale of 2π-periodic functions, or the scale of 2π-antiperiodic functions, or the scale of 4π-periodic ones. It is well-known [Ma, MT] that the spectrum of the Sturm–Liouville operator Lu acting on twice differentiable functions of period 4π is a sequence of simple or double eigenvalues {λj | j ≥ 0}, tending to infinity:

λ0 < λ1 ≤ λ2 < λ3 ≤ λ4 < · · · & ∞. Corresponding eigenfunctions are smooth if the potential u(x) is. The spectrum {λj } can be also described without doubling the period: it equals the union of the periodic and antiperiodic spectra of the operator Lu , considered on the segment [0, 2π]. Below we denote λ = {λ0 , λ1 , . . . } and refer to the sequence λ = λ(u) as to the periodic/antiperiodic spectrum of the operator Lu . The segment ∆j = [λ2j−1 , λ2j ], j = 1, 2, . . . , is called the j th spectral gap. The gap ∆j is open if λ2j > λ2j−1 and is closed if λ2j = λ2j−1 . Example. For u = 0 we have λ2k = k 2 /4, k ≥ 0, and λ2l−1 = l2 /4, l ≥ 1. Corresponding eigen-functions are (2π)−1/2 cos kx/2 and (2π)−1/2 sin lx/2. All gaps ∆j are now closed. If u(t, x) is a smooth x-periodic function, then the linear equation v˙ = Au(t,x) v, v(0, x) = v0 (x), has a unique smooth periodic solution v(t, x) for any given smooth periodic initial data v0 (x) (this follows from an abstract theorem in [Paz]). Hence, Lemma 4 with {Zs = H s (R/4πZ)} imply that the sequence λ is an integral of motion: λ(u(t, ·)) is time-independent if u(t, x) is a solution of the KdV.

(4.1)


81

Let us fix any integer n-vector Υ = (V1 , . . . , Vn ) ∈ Nn , where V1 < · · · < Vn , and consider a set TΥ2n , TΥ2n = {u(x) | the gap ∆j (u) is open iff j ∈ {V1 , . . . , Vn }}. n This set equals to the union of isospectral " subsets T n (r) = TΥ (r) with 2n n n prescribed lengths of the open gaps: TΥ = r∈Rn TΥ (r), where TΥ (r) = +

n (r) is invariant for the {u(x) ∈ TΥ2n | |∆Vj | = rj ∀ j}. By (4.1) each set TΥ KdV-flow. Remarkably, the whole spectrum λ of an n-gap potential is defined by n the n-vector r and analytically depends on it [GT]. Each set TΥ (r) is not n empty and is an analytic n-torus in any space H0s = H0s (S 1 ). The tori TΥ (r) are analytically glued together, so TΥ2n is an analytic submanifold of each space H0s . – These are well-known results from the inverse spectral theory of the Sturm-Liouville operator Lu , see [Ma] and [Mo, GT, MT]. So the inverse spectral theory provides us with KdV-invariant 2n-manifolds foliated to invariant n-tori.

4.2

The Its–Matveev theta-formulas

To check that the n-gap manifolds TΥ2n of the KdV equation possesses the properties (i)–(iv) from Section 2, we have to present an analytic map Φ0 as in Section 2 and to check its properties. We shall write the map Φ0 in terms of theta-functions, following [D, BB]. n Let us take any n-gap potential u(x) ∈ TΥ (r) and denote by E1 (r) < E2 (r) < · · · < E2n+1 end points of the open gaps plus λ0 (so E1 = λ0 and ∆V1 = [E2 , E3 ], . . . , ∆Vn = [E2n , E2n+1 ]). The Riemann surface Γ = Γ(r) of genus n, Γ = {P = (λ, µ) | µ2 = R(λ; r) :=

2n+1

(λ − Ej (r))},

j=1

has branching points at E1 , . . . , E2n+1 and ∞. Let a1 , . . . , an be the ovals in Γ lying above the open gaps ∆V1 , . . . , ∆Vn (i.e., aj = π −1 ∆Vj ). After Γ is cut along these ovals, it falls to two sheets Γ+ , Γ− , chosen in such a way that µ is positive on the upper edge of the cut [E2n+1 , E∞ ] in Γ+ . We supplement the ovals aj by n b-circles b1 , . . . , bn in such a way that the circles have the canonical intersection matrix: ai ◦ aj = bi ◦ bj = 0 and ai ◦ bj = δij . Next we take a basis dω1 , . . . , dωn of #holomorphic differentials on Γ, normalised by the conditions dωj , ak := ak dωj = 2πiδjk . These differentials exist and are uniquely defined by the normalisation. The Riemann matrix B = B(r) = (Bjk ) of the curve Γ is defined as the matrix of b-periods of the differentials dωj : Bjk = dωj , bk .

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Under our choice of the a, b-cycles, the matrix B is real (see [BB, KK]). Its symmetric part is negatively defined due to general properties of the Riemann matrices. Now we define the theta-function θ of the curve Γ = Γ(r): 1 (B(r)s, s) + (z, s) , exp θ = θ(z; r) = 2 n s∈Z

where z ∈ C (the sum converges due to the properties of the Riemann matrix B). Clearly the function is 2π-periodic in imaginary directions: θ(z+2πiek ) = θ(z), where ek is the k-th basis vector of Cn . The differentials dωj analytically depend on the parameter r ∈ Rn+ as well as the matrix B(r). Therefore the function θ(z; r) is analytic in r ∈ Rn+ . Since the matrix B is real, then θ is real and even: θ(z) = θ(z) and θ(z) = θ(−z). In particular, this function is real both in real and pure imaginary directions. Next, on the surface Γ(r) we consider Abelian differentials of the second kind dΩ1 , dΩ3 with vanishing a-periods and with the only poles at infinity of the form √ dΩ1 = dk + (c + O(k−2 )) dk −1 , k = i λ → ∞, (4.2) dΩ3 = dk 3 + O(1) dk −1 , n

where c is an unknown constant. The normalisation (4.2) defines the differentials uniquely, see [S, NMP,BB]. Besides, dΩ1 =

i λn + . . . 3 λn+1 + . . . dλ, dΩ3 = − i dλ, 2 µ 2 µ

(4.2 )

where the dots stand for real polynomials of degree n − 1. Let us define complex n-vectors iV(r) and iW(r) as the vectors of bperiods of these differentials: iVj = dΩ1 , bj ,

iWj = dΩ3 , bj .

The vector V is called the wave-number vector and W – the frequency vector. The vector W is real and the vector V is integer. Moreover, V equals to the vector, formed by the numbers of the open gaps, see [BB, KK] (the latter vector was earlier denoted also as V ). One of top achievements of the finite-gap theory is the Its–Matveev formula, which represents any n-gap potential u(x) ∈ T n (r) in the form u(x) = u(x; r, z) = 2

∂2 ln θ(iVx + iz; r) + 2c. ∂x2

(4.3)

Here the constant c is the same as in (4.2). Since the mean-value of the r.h.s. in (4.3) equals 2c, then we must have c = 0. The vector iz in (4.3)


83

is iz = −A(D) − K , where K is the vector of Riemann constants (see [D, BB]) and A(D) is the Abel transformation of a positive divisor D = D(), D = D∞ . . . D\ , Dj ∈ aj . I.e., A(D) is a complex n-vector such that its jth component A(D)| equals A(D)j =

n r=1

Dr

∞

dωj ,

where {dωj } are the holomorphic differentials on Γ as above. The divisor D is a divisor of Dirichlet eigenvalues, i.e. Dj = (λj , µj ), where λj is an eigenvalue of the operator Lu subject to Dirichlet boundary conditions ϕ(0) = ϕ(2π) = 0 (each gap ∆j contains exactly one point from the Dirichlet spectrum, see [Ma, MT]).7 In particular, every point Dj analytically depends on u. The phase vector z turns out to be real, so θ(iVx + iz) is a real valued function of x. The theta-function is nonzero at any imaginary point iξ ∈ iRn (see in [BB]). Since this function is periodic, then |θ(iξ)| ≥ C(r) > 0 for any real vector ξ. Hence, the r.h.s. of (4.3) is analytic in z ∈ T n . Due to the periodicity, we can treat z as a point in the torus T n . Thus we get an analytic map: T n (r) → T n , u(·) → z. This map has the analytic inverse given by the formula (4.3). The coordinate z on a finite-gap torus T n (r) is called the theta-angles. Time-evolution u(t, x) of the n-gap potential u(x) ∈ T n (r) as in (4.3) along the KdV flow is given by the following formula, also due to Its–Matveev: u(t, x; r, z) = 2

∂2 ln θ(i(Vx + Wt + z); r) ∂x2

(4.4)

(we use that c = 0). ∂2 Denoting by G the function G(z, r) = 2 ∂x 2 ln θ(iVx + iz; r) |x=0 , we write u as u(x; r, z) = G(z + V x, r). So the n-gap torus T n (r) is represented in the form T n (r) = Φ0 (r, T n ) ⊂ H0d , where for any z ∈ T n , Φ0 (r, z) is the analytic function Φ0 (r, z)(x) = G(Vx + z, r). The formula (4.4) can be written as u(t, x; r, z) = Φ0 (r, z + W(r)t)(x). This shows that in the (r, z)-variables the KdV-flow on T 2n takes the form z˙ = W(r). I.e., the theta-angles z integrate the KdV-equation on any torus T n (r). 7

This divisor can be also described as a divisor of poles of the Baker–Akhiezer eigenfunction ϕ(x; P ) of the operator Lu , Lu ϕ = π(P )ϕ, normalised at infinity √ as ϕ ∼ ei λ x ; see [D, BB].

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Let R be a sub-cube of the octant Rn+ of the form R = {r ∈ Rn+ | 0 < rj < K} with some K > 0, and T 2n = Φ0 (R × T n ) ⊂ TV2n

(4.5)

for any wave-number vector V. The set T 2n ⊂ H0d , d ≥ 1, is an invariant manifold of the KdV equation. It meets the assumptions i)–iii) from Section 2 since: The map Φ0 is an analytic embedding and T 2n is an analytic submanifold of H0d . The form Φ∗0 α2 is analytic and is non-degenerate for small r since it is nondegenerate at r = 0 by Theorem 3.1, so the set of its degeneracy is a proper analytic subset of the cube R (in fact, it is empty). The non-degeneracy assumption iv) also holds, as states the following Non Degeneracy Lemma: Lemma 6. The determinant det{∂Wj /∂rk } is nonzero almost everywhere. This result can be proven directly [BiK2, KK], or can be obtained as an immediate consequence of another lemma (proven in [BoK1, KK]): Lemma 7. For any finite-gap manifold TV2n the corresponding frequency vector W as a function of r analytically extends to the origin and has there the 3 following asymptotic: Wj (r) = − 14 Vj3 + 8πV r2 + O(|r|4 ), j = 1, . . . , n. j j 4.3

Higher equations from the KdV hierarchy

Let us take any n-gap manifold TV2n . The manifold itself and each torus T n (r) ⊂ TV2n are invariant for all Hamiltonian equations with the hamiltonians H0 , H1 , . . . from the KdV-hierarchy (see Example 3.2). The flow of any l-th KdV equation on TV2n is very similar to the KdV-flow: it is given by the theta-formula (4.4) where the frequency-vector W should be replaced by an n-vector W(l) , formed by b-periods of some Abelian differential dΩ2l+1 . All results till Lemma 6, stated above for the KdV equation, have obvious reformulations for the higher KdV’s. An analogy for Lemma 7 is given by the following statement: The vector W(l) is analytic in r12 , . . . , rn2 and (l)

(l)

(l)

Wj (r) = Wj0 + Wj1 r2 + O(|r|4 )

(4.6)

(l)

for any j = 1, . . . , n, with some non-zero constants Wj1 . Any manifold TV2n treated as an invariant manifold of an lth KdV equation satisfies assumptions i)-iv) for the same reason as for l = 1 (i.e., as in the KdV-case). 4.4

Sine-Gordon equation under Dirichlet boundary conditions

Similar to the KdV-case, the equation (SG)+(EP) (see Example 3.3) has time-quasiperiodic finite-gap solutions with n basic freqencies which can be


85

written in terms of theta-functions. To symplify presentation we restrict ourselves to finite-gap solutions with even number of gaps, so g = 2n everywhere below. These solutions can be written as ξ = (u, v)(t, x; D, r), where v = −u˙ and u(t, x) = 2i log

θ(i(V x + W t + D + ∆)) . θ(i(V x + W t + D))

(4.7)

Here θ is a theta-function of the Riemann surface Γ, Γ = Γ(r) = {(λ, µ) | µ2 = λ

n

¯j )(λ − E −1 )(λ − E ¯ −1 )}, (λ − Ej )(λ − E j j

j=1

∆ is the vector (π, . . . , π), the wave-number vector V and the frequency vector W are constructed in a way, similar to the KdV-case and D = (D1 , . . . , Dn , Dn , Dn−1 , . . . , D1 ),

D = (D1 , . . . , Dn ) ∈ Tn .

The vector r = (E1 , . . . , En ) is a point from some connected n-dimensional real algebraic set R, which is a bounded part of the set Cn+ , C+ = {x + iy | y > 0}. The algebraic set R is “smooth near the real subspace” in the sence that for some δ0 > 0 the set R+ = {(E1 , . . . , En ) ∈ R | 0 < Ej < δ0 ∀ j} is a smooth n-dimensional real analytic manifold. Since the set of branching points of the surface Γ is inversion-invariant, then the vector W (the vector V ) turned out to be symmetric (antisymmetric) with respect to the involution T , T (U1 , . . . , U2n ) = (U2n , . . . , U1 ): TW = W ,

T V = −V .

˜ and V˜ , formed by the Hence, W and V are determined by the vectors W last n components: ˜ = (Wn+1 , . . . , W2n ) , W

V˜ = (Vn+1 , . . . , V2n ).

These vectors also will be called the frequency and wave-number vectors. The wave-number vector V˜ (r) is an r-independent integer vector (this guarantees that u is 2π-periodic in x). An important property of the family ¯ contains an unique real point r0 = of solutions we discuss is that the closure R 0 0 2n (E1 , . . . , En ) ∈ R and u → 0 when r → r0 (this convergence corresponds ¯j ] shrink to to a degeneration of the Riemann surface Γ when the gaps [Ej , E points). Moreover, u(0, x; D, r) =

n

! εj cos(Vn+j x + Dj ) + cos(−Vn+j x + Dj ) + o(|ε|2 ),

j=1

(4.8)

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where ε → 0 as r → r0 . For the sake of simplicity we restict ourselves to finite-gap solutions such that first n gaps are open in the sence that (Vn+1 , . . . , V2n ) = (1, . . . , n).

(4.9)

˜ is analytic on the algebraic set R and satisfies The frequency vector W the following analogies of Lemmas 6 and 7: ˜ (r)∗ is nondegenerate for a.a. r ∈ R; a) the tangent map W b) the set R+ admits analytic coordinates µ1 , . . . , µn such that Im Ej → 0 ˜ is analytic in µ and as µj → 0, the map W Wn+j (0) = 1 + j 2 =: j ∗ ,

(4.10) −16/j ∗ , j ≤ n, j = k, ∂Wn+j |µ=0 = ∗ ∂µk −12/j , j = k. The set of time-quasiperiodic solutions (4.7) is incomplete in the sence that the infinitesimal solutions (4.8) do not contain the “cos 0x” term ε0 · 1. The reminding solutions are odd-gap, they can be treated similar. For all these results see [BB], [BiK1, BoK2] and [KK]. A solution (4.7) is winding in a torus T n (r) = {ξ(0, ·; D, r) | D ∈ Tn }. " Altogether the tori form a set T 2n = r∈R T n (r) which is an image of Tn ×R under a map Φ0 , defined in terms of the r.h.s. of (4.7) as in the KdV-case. The set T 2n and the map Φ0 satisfy the assumptions (i)–(iii) for the same trivial reasons as in the KdV-case, while assumption iv) follows from a).

5 5.1

Linearised equations and their Floquet solutions The linearised equation

Below (Z, α2 ) stands for a symplectic space (Z = Zd , α2 = J dz ∧ dz) with some fixed d. We continue to study a quasilinear Hamiltonian equation u˙ = J∇H(u) = J(Au + ∇H(u)) =: VH (u),

(5.1)

where ord A = dA , ord ∇H = dH < dA , ord J = dJ and d ≥ dA /2. The equation is assumed to possess a 2n-dimensional invariant manifold T 2n ⊂ Z as in Section 3, satisfying the assumptions i)-iv). By iii), the regular part T02n of T 2n is filled by smooth solutions u0 (t) of (5.1) of the form u0 (t) = u0 (t; r0 , z0 ) = Φ0 (w0 (t)), where w0 (t) = (r0 , z0 + tω(r0 )) ∈ R0 × T n , t ∈ R. We linearise (5.1) about a solution u0 as above to get the nonautonomous linear equation v˙ = J(Av + ∇H(u0 (t))∗ v) =: JAt (t)v,

(5.2)


87

which is our concern in this section. We recall that linear flow-maps of equation (5.2) (if they exist) are denoted as Sτt ∗ (u0 (τ )) (see Definition 1), and supplement the assumptions i)–iv) assuming that (v) for any solution u0 of (5.1) in T02n the flow-maps Sτt ∗ (u0 (τ )), −∞ < τ, t < ∞, are well defined in the space Z = Zd . Since the equation (5.1) is Hamiltonian, then the flow-maps Sτt ∗ (u) (u ∈ T02n ) are symplectomorphisms of the symplectic space (Z, α2 ) (see [KK]). To study equation (5.1) near T 2n we shall impose an integrability assumption on the linearised equation (5.2). Roughly speaking, this assumption means that the equation (5.2) has a complete system of time-quasiperiodic Floquet solutions. Since the time-flow of (5.2) is formed by linear symplectomorphisms which preserve tangent spaces to T02n , then the flow also defines symplectomorphisms of skew-orthogonal complements Tu⊥0 T02n to the spaces Tu0 T02n in Tu0 Z ∼ Z.8 5.2

Floquet solutions

We call a solution v(t) of the equation (5.2) a Floquet solution if there exists a section Ψ of the complexified tangent bundle to Z, restricted to T02n (i.e., R0 × T n (r, z) → Ψ(r, z) ∈ TΦ0 (r,z) T02n ), and a complex function ν(r) such that the solution v has the form v(t) = v(t; r0 , z0 ) = eiν(r0 )t Ψ(w0 (t)),

w0 = (r0 , z0 + tω(r0 )).

(5.3)

It is assumed that v(t) solves (5.2) for any choice of r0 ∈ R0 and z0 ∈ Tn . We call ν(r) the (Floquet) exponent of a Floquet solution v. A Floquet solution v(t) is called a skew-orthogonal Floquet solution if Ψ in (5.3) is a section of the complexified skew-normal bundle T ⊥c T 2n (its fibres are complexifications of the spaces Tu⊥0 T02n ). Let us assume that equation (5.2) has an infinite family of Floquet solutions v = vj (t). Clearly if vj is a Floquet solution, then v j is a solution with the exponent −ν j (r), corresponding to the section Ψj . We add this solution to the family; if a solution with the exponent −ν(r) already was there, we replace it by v j . Now the family is invariant with respect to the complex conjugation and the set of all exponents is invariant with respect to the involution ν → −ν. In addition we suppose that the set of exponents is invariant with respect to the complex conjugation ν → ν (this assumption holds trivially if all the frequencies are real); hence the set is invariant with respect to the involution ν → −ν. It is convenient to enumerate the Floquet solutions by integers from the set Zn = {±(n + 1), ±(n + 2), . . . }. We do it in such a way that ν−j (p) ≡ −νj (p) 8

Tu⊥0 T02n is formed by all vectors ξ ∈ Tu0 Z such that α2 (ξ, η) = 0 for each η ∈ Tu0 T02n .

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and Ψ−j ≡ Ψj if νj is real. So below we consider the following system of Floquet solutions : vj (t; r0 , z0 ) = eiνj (r0 )t Ψj (r0 , z0 + tω(r0 )), j ∈ Zn ; ν−j (r) ≡ −νj (r). (5.4) ˆ For each index k we denote by kˆ an index such that νkˆ = νk . Clearly kˆ = k for any k and kˆ = k if νk is real. We note that the hat-map is r-independent in any connected sub-domain of R0 where all the functions νj (r) are different. A frequency νj can have an algebraic singularity at the set Λj formed by all r such that νj (r) = νp (r) for some p = j. Situation becomes too intricate if there are infinitely many nontrivial sets Λj . To avoid this complexification we assume that a) there is a nonempty sub-domain of R0 where νj = νk if j = k. Moreover, there exists j1 (depending on T 2n ) such that νj (r) = νk (r) for all r, all k and all |j| ≥ j1 , j = k. Since ν−j = −νj , then by this assumption νj = 0 if j ≥ j1 . The exponents νk with |k| ≥ j1 are assumed to be real analytic: b) for any k such that |k| ≥ j1 , νk is a real-valued analytic function on R (so νk ≡ −ν−k and Ψ−k = Ψk ). The section Ψk extends to an analytic map Πc × {| Im z| < δ} → Z c and νk extends to an analytic function on Πc . In particular, kˆ = k if |k| ≥ j1 . For sophisticated integrable equation like the SG equation some exponents νk (r) with |k| < j1 have non-trivial algebraic singularities. The assumptions we shall impose now on the exponents νk with |k| < j1 (below we call such k small ), are made ad hoc – they are met by Floquet solutions of Laxintegrable equations. These assumptions are empty for integrable equations with seladjoint L-operators, e.g., for the KdV equation. Let us denote M = j1 − n − 1. c) The functions νj with small j are continuous in R and have the form νj (r) = ν˜(λj (r), r), where {λj (r)} are branches of some 2M -valued algebraic function, defined on Πc , and ν˜ is an analytic complex function. By c) the functions νj with small j are algebraic (as well as the functions λj ). They are analytic in Πc outside a discriminant set D of the algebraical function ν˜. We note that D ∩ R is a proper analytic subset of R since no two exponents νj , νk coincide identically in R by the assumption i). The set D ∩ Rc contains algebraic singularities of the Floquet exponents and is contained in the set Λ = {r ∈ Rc | νj (r) = νk (r) for some j = k, with |j|, |k| < j1 }, which is a proper analytic subset of Rc . It contains zeroes of the exponents νj since they are odd in j. We add Λ to the singular set Rsc : Rsc := Rsc ∪ (Λ ∩ Rc ),

Rs := Rs ∪ (Λ ∩ R) ,


89

and modify the regular set R0 accordingly. Example. Eigenvalues {λj } of a real matrix B(a) which analytically depends on a real vector-parameter a are zeroes of the characteristic equation det(B(a) − λE) = 0 and are algebraic functions of a. A priori they have singularities at the sets Λjk = {λj = λk }. Some of these singularities can be removed by re-enumerating the eigenvalues before or behind the sets Λjk . In particular, if the matrix B(a) is symmetric, then under proper enumeration the eigenvalues have no singularities at all (this is Rellich’s theorem, see [Kat2,RS4]). However, if λj and λk are real “before” Λjk and have nontrivial imaginary parts “behind” Λjk , then a singularity at this set is unremovable. Now we pass to smoothness of the sections Ψj with small j (|j| < j1 ): $ z, λ) ∈ Z c , such that d) There exists a bounded analytic map (r, z, λ) → Ψ(r, n $ Ψj (r, z) = Ψ(r, z; λj (r)) for (r, z) ∈ R0 × T and all small j. This assumption agrees with smoothness of eigenvectors in finite-dimensional spectral problems: j Example (continuation). Let us denote by B j = (Blm | 1 ≤ l, m ≤ n) the matrix B j (a) = B − λj (a)E, so " Bξ = λj ξ if B j ξ = 0. Let us assume that rk B j (a) = n − 1 for a ∈ / Λj = k Λjk and denote by ξm (a) the algebraic j j (a) in complement to the element Bnm the jmatrix B . Then the vector ξ = (ξ1 , . . . , ξn ) is nonzero for a ∈ / Λj and m Blm ξm = 0 since: for l = n the sum equals det B j = 0 and for l = n it vanishes by an elementary linear algebra. The vector ξ is an eigenvector of B. It is a polynomial in the eigenvalues λj and in the elements of the matrix B. It vanishes at Λj .

5.3

Complete systems of Floquet solutions

Let us take any basis {ϕj | j ∈ Z0 } of the Hilbert scale {Zs } and assume that the basis is symplectic, i.e., α2 [ϕj , ϕ−k ] = Jϕj , ϕ−k =

δj,k νjJ

for all j, k ∈ Z0 ,

(5.5)

J and νjJ > 0 if j is positive. Due to (5.5), Jϕk = ϕ−k /νkJ where νjJ = −ν−j for any k ∈ Z0 . Since J is an anti selfadjoint isomorphism of the scale {Zs } of order −dJ ≤ 0, then C1−1 j dJ ≤ νjJ ≤ C1 j dJ for every j ≥ 1 with some C1 ≥ 1. For any real s we denote by Ys the Hilbert space

Ys = span{ϕj | j ∈ Zn } ⊂ Zs . The spaces {Ys , α2 |Ys } form a symplectic Hilbert scale with the basis {ϕj | j ∈ Zn }. In Y0c we choose some complex basis {ψj | j ∈ Zn } and assume that it is symplectic: α2 [ψj , ψ−k ] = δj,k µj .

(5.6)

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Besides, we assume that for big j it agrees with the basis {ϕj }: √ µj = i/νjJ and ψ±j = (ϕj ∓ iϕ−j )/ 2 if j ≥ j1 .

(5.7)

Example. If {ϕj } is the trigonometric basis as in (1.1), i.e. ϕk = π −1/2 cos kx and ϕ−k = −π −1/2 sin kx, then for |k| ≥ j1 the functions ψk are complex exponents, ψk = (2π)−1/2 eikx . Let {vj } be a system of Floquet solutions as in Section 5.2 and {Ψj } are corresponding sections. For any (r, z) ∈ R0 ×T n we denote by Φ1 (r, z) a linear map from Y c = Ydc to Z c which identifies ψj with Ψj : Φ1 (r, z) : Y c → Z c ,

ψj → Ψj (r, z) ∀j ∈ Zn .

(5.8)

The map Φ1 will be used to formulate an important assumption of completeness of a system of Floquet solutions. Before to do this we cut out of the set R0 some “neighbourhood of infinity” and any neighbourhood of the singular set Rs to get an open domain R1 , R1 R0 \ Rs . Possibly, R1 is disconnected. To simplify notations we assume that R1 belongs to a single chart of the analytic manifold R0 and treat R1 as a bounded domain in Rn . We fix any bounded complex domain R1c which contains R1 with its δ-neighbourhood and does not intersect the singular set Rsc . We denote by W1 the set W1 = R1 × T n and denote by W1c its complex neighbourhood, W1c = R1c × {|Im z| < δ}. Definition 3. A system of Floquet solutions (5.4) which satisfies the analyticity assumptions a)–d) is called complete (in the space Z = Zd ) if: 0) it is formed by skew-orthogonal Floquet solutions, and for any (r, z) ∈ R0 × T n we have: 1a) the functions βj = iα2 [Ψj (r, z), Ψ−j (r, z)], j ∈ Zn , are z-independent: βj = βj (r), b) there is a non-empty sub-domain of R where no function βj (r) vanishes identically, c) the vectors {Ψj (r, z)} form a skew-orthogonal system in the space TΦ⊥c T 2n , that is: 0 (r,z) α2 [Ψj , Ψ−k ] = iβj (r)δj,k

∀j, k .

(5.9)

2) The vectors {Ψj (r, z)} are uniformly asymptotically close to the complex basis {ψj } and the exponents νj (r) are close to constants. Namely, a) the linear map Φ1 (r, z) equals to the natural embedding ι Y → Z up to a ∆-smoothing operator, ∆ > 0: Φ1 (r, z) − ιd,d+∆ ≤ C1

for all (r, z) ∈ W1c ;

(5.10)


91

b) for large j the functions βj (r) in (5.9) are analytic in R1c and are there close to the constants (νjJ )−1 , defined in (5.5) (cf. (5.7)): |βj (r) − 1/νjJ | ≤ C2 |j|−κ

for r ∈ R1c

(5.11)

with some κ ≥ max(1, dJ + ∆); c) the exponents νj are bounded in R1c and are there “asymptotically close to constants”. That is, |νj (r)| ≤ C3 |j|dA +dJ for r ∈ R1c and

|∇νj (r)| ≤ C4 |j|∆

for r ∈ R1c

(5.12)

$ < d A + dJ . with some real ∆ The constants C1 − C4 in this definition may depend on the domain R1 but not on j. Since Ψ−j (w) = Ψj (w) for real w and big j, then the corresponding functions βj are real and β−j ≡ −βj . As µj ≥ C −1 j −dJ , then by the assumption (5.11) we have that |βj (r)| ≥ 12 µj for all r ∈ R1c and j > j2 , with some new constant j2 . We consider the product $b(r) =

j2

βj2 (r).

j=n+1

It follows from the properties c) and d) that the function ˜b is analytic in Πc (see [KK]). Due to 1b) a set of its zeroes in Rc is a proper analytic subset. We add it to the complex singular set Rsc and accordingly modify the sets Rs and R0 . If it is necessary, we also decrease the domain R1 so that the inclusion R1 R0 \ Rs still holds true. Remark. The set Rs as it is defined now is the final singular set for our constructions. It comprises: 1) the singular part of the algebraic set R, 2) the set of degeneracy of the pull-back symplectic structure Φ∗0 α2 , 3) algebraic singularities of the Floquet exponents and 4) points where any two of them coincide. Finally, it contains 5) the zero-set of the function ˜b we have just constructed. The last set is a set of degeneracy of the system {Ψj } since a vector Ψj (r, z) is skew-orthogonal to the tangent space Tu T 2n and to all the vectors Ψk (r, z) as soon as βj (r) = 0. In the same time in Lemma 8 below we prove that the vectors {Ψj } form a basis of the skew-orthogonal space Tu⊥c T 2n if r ∈ / Rs . ˜ \ Rs may be chosen to occupy most part of R0 in the The set R1 R ˜ is a bounded chart of the manifold R0 , mesn is the sense of measure: If R ˜ and γ is any positive number, then R1 can be chosen Lebesgue measure in R in such a way that ˜ \ R) ≤ γ. mesn (R

(5.13)

A complete system of skew-orthogonal Floquet solutions spans the skeworthogonal spaces Tu⊥c T 2n in conformity with the term “complete” we use:

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Lemma 8. For any w ∈ W1c the map Φ1 (w) defines an isomorphism of the spaces Y c and Tu⊥c T 2n , where u = Φ0 (w). 9 Proof. By (5.10) the map Φ1 (w) is a compact perturbation of the embedding ι Y c → Z c , so ind C Φ1 (w) = ind ι = 2n. As a range of Φ1 lies in Tu⊥c T 2n , then dimC Coker Φ1 ≥ 2n. So if we can show that Ker Φ1 = {0}, then the range of Φ1 equals Tu⊥c T 2n and the assertion will follow. Suppose that the ψ kernel is non-trivial. Then it contains a nonzero vector ξ = y j j and we have 0 = Φ1 ξ = yj Ψj (w). The skew-inner product of the right-hand side with any vector Ψ−j (w) equals i yj βj (r) (see (5.9)). Thus, yj ≡ 0 and ξ = 0. Contradiction. Decreasing in a need the complex neighbourhood W1c of W1 we easily get the following result: Lemma 9. For any s ∈ [−d − dJ − ∆, d + ∆] the operator Φ1 (w) : Ysc → Zsc analytically depends on w ∈ W1c and is uniformly bounded. Moreover, for any c is analytic in w ∈ W1c as well. s as above the map Φ1 (w) − ι Ysc → Zs+∆ Example ((Birkhoff-integrable systems, see [K1, p. 401). and [Kap, BKM])] Let Z be a space of sequences ξ = (x1 , y1 ; x2 , y2 ; . . . ), given some Hilbert norm and given the “usual” symplectic structure by means of the 2-form J dξ ∧ dξ, where J(x1 , y1 ; . . . ) = (−y1 , x1 ; . . . ). We denote pj = (x2j + yj2 )/2, qj = Arg(xj + iyj ) and consider an analytic hamiltonian h(p1 , p2 , . . . ). The subspace T 2n ⊂ Z formed by all vectors ξ such that 0 = xn+1 = yn+1 = . . . is invariant for the Hamiltonian vector field Vh and the restricted to T 2n system is obviously integrable. Let us denote pn = (p1 , . . . , pn ), q n (t) = (q1 , . . . , qn ) n ) and denote by νj the functions νj (pn ) = ∂h(p∂p,0,... , j ≥ 1. We shall identify j n n p with the vector (p , 0, . . . ). The manifold T 2n is filled with solutions ξ(t) = {pn = const , q n = tν n (pn ) + ϕn ; pr = 0 for r > n}, where ϕn ∈ T n and ν n = (ν1 , . . . , νn ). For any j > n let us consider a smooth variation ξ(t, ε) of a solution ξ(t), which changes no action pl except pj and makes the latter equal ε2 : ξ(t, ε) = {pn (t) = pn , q n (t) = tν n (pn )+q0n (ε); xl (t) = yl (t) = 0 if l > n, l = j} and xj (t) = ε cos(tνj (pn )+ϕ(ε)), yj = ε sin(tνj (pn )+ϕ(ε)) , where ϕ(ε) ∈ S 1 . ∂ ξ(t, ε) |ε=0 is a solution of the equation, linearised about The curve v˜j = ∂ε ξ(t). It equals v˜j (t, ϕ) = {δpn = 0, δq n = q0n (0); δx(t), δy(t)}, 9

For a complex w and u = Φ0 (w) we define Tu⊥c T 2n as the set of all z ∈ Z c such that α2 [z, Φ0 (w)∗ ξ] = 0 for any ξ ∈ Tw W1c .


93

where δxr (t) = δyr (t) = 0 if r = j and δxj = cos(tνj (pn ) + ϕ), δyj = sin(tνj (pn ) + ϕ),

ϕ = ϕ(0).

The curve vtriv (t) = {δpn = 0, δq n = q0n (0); δx = δy = 0} is a trivial solution of the linearised equation (it may be obtained using a variation of ξ(t), corresponding to a rotation of the phase-vector q n ). An appropriate complex linear combination of the solutions v˜j (t, 0), v˜j (t, π) and of the trivial n solution as above takes the Floquet form vj (t) = eiνj (p )t Ψj , where Ψj = (0, . . . ; i, 1; 0, . . . ) (the pair (i, 1) stands on the jth place). Let us suppose that |νj | ≤ Cj dA for some dA and that (5.12) holds. Then the system of Floquet solutions {vj , vj | j ≥ n + 1} is complete in the sense of Definition 3. This example illustrates well the definition but it is too simple and so too restrictive: to be Birkhoff integrable a finite-dimensional system has to have dim Z/2 integrals of motion, but to have a complete system of Floquet solutions for the equations linearised about solutions in T 2n it needs only n of them (see below Proposition 2). To be useful in analytical studies of the equation (5.1) and its perturbations, a system of Floquet solutions should be complete and non-resonant: Definition 4. A system of Floquet exponents {νj (r) | j ∈ Zn } satisfying a)–c) is called non-resonant if: 3) there exists a domain O ⊂ R0 such that for all s ∈ Zn and all j, k ∈ Zn , j = −k, we have: ω(r) · s + νj (r) ≡ 0 in O, ω(r) · s + νj (r) + νk (r) ≡ 0 in O.

(5.14) (5.15)

The system of Floquet solutions with non-resonant exponents also is called non-resonant. Since the exponents νj are algebraic (or analytic) functions, then zero-set of any resonance is nowhere dense. Finally we give: Definition 5. A system of Floquet solutions (5.4) satisfying a)–d) is called complete non-resonant if it satisfies assumptions 0)–3) from Definitions 3, 4. It turns out that the assumption 1) follows from 3): Lemma 10. Any non-resonant system of Floquet solutions satisfy assumptions 0),1a) and 1c) from Definition 3. Proof. To check 1c) we should prove that for any j = −k the function F (r, z) = α2 [Ψj , Ψk ] vanishes identically. To do it let us consider the function f (t; r, z), f := ei(νj +νk )t α2 [Ψj (w(t)), Ψk (w(t))] = α2 [vj (t), vk (t)],

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where w(t) = (r, z + tω(r)). Since the flow-maps Stt12∗ are symplectic [KK], then the skew-product of any Floquet solutions vj and vk is time-independent and df %% = i(νj + νk )F + z F · ω. 0= % dt t=0 Let us expand F as Fourier series, F = eis·z F&(r, s). From the last identity we get that F&(r, s)(νj + νk + s · ω(r)) = 0 for all s and all r. Since the second factor is nonzero for almost all r, so F&(r, s) ≡ 0 and F (r, z) ≡ 0. If j = −k, then we have: const ≡ α2 [vj (t), v−j (t)] = α2 [Ψj (w(t)), Ψ−j (w(t))]. Because the assumption iv) (Section 3), the curve w(t) is dense in a torus {r} × Tn for almost all r. So the left-hand side of (5.9) with k = −j is z-independent for almost all r. By continuity, it is z-independent for all r, as states 1a). To check 0) one has to argue similar, using variations δr, δz of the initial conditions for the curve w(t). Corollary. A system of Floquet solutions (5.4) which meets the assumptions a)–d) from Section 5 as well as the assumptions 2), 3) from Definitions 3, 4 is skew-orthogonal to T 2n and is complete non-resonant, provided the assumption 1b) holds. The latter happens e.g., if there exists a point r∗ ∈ R such that Ψj (r, z) → ψj as r tends to r∗ . Here R signifies the closure of R in RN where R is a subset. Practically the point r∗ corresponds to the zero-solution of the equation (5.1) (or another trivial solution). This result simplifies verification of completeness for a system of Floquet solutions since it is much easier to check the non-resonance relations (5.14), (5.15) than the completeness 1a)–1c). The transformation Φ1 integrates the linearised equation (5.2): it sends the curves yj = eiνj (r0 )t ψj to solutions vj (t) of (5.2). It is convenient to have this transformations symplectic. For this end the sections {Ψj } have to be properly reordered and normalised by multiplying by some analytic functions of r; simultaneously the basis {ψj } also have to be transformed by a linear symplectomorphism which changes finitely many its components only. In this way the following result can be proven: Proposition 2. Given any complete system of Floquet solutions (5.4) we can normalise the sections {Ψj } and the basis {ψj } is such a way that the new basis still meets (5.6), (5.7) and the new system of Floquet solutions still is complete. Besides, a) for any (r, z) ∈ W1 = R1 × Tn the map Φ1 (r, z) defines a symplectic ⊥ T 2n which analytically in (r, z) ∈ W1c extends isomorphism of Y and TΦ(r,z) c to a bounded linear map Y → Z c ; b) the nonautonomous linear map Φ1 (r, z+tω(r)) sends solutions of an autonomous equation y˙ = JB(r)y, y ∈ Y c , to solutions of (5.2). The selfadjoint


95

operator B(r) is analytic in r. Besides, ord B(r) ≤ dA and ord ∇r B(r) ≤ $ Spectrum of the operator JB(r) equals to the set of Floquet expo−dJ − ∆. nents of the solutions (5.4). The basis {ψj } may depend on a connected component of the set R1 . The leading Lyapunov exponent of equation (5.2) in Zd is a number a equal to the supremum over all real numbers a such that lim t→∞ e−a t v(t)d = ∞ for some solution v(t) ⊂ Zd of (5.2). A solution u0 (t) of (5.1) is called linearly stable if the leading Lyapunov exponent of the corresponding linearised equation (5.2) vanishes. A direct consequence of Proposition 2 is the following Corollary. If the linearised equations (5.2) have complete system of Floquet solutions, then the leading Lyapunov exponent of the equation corresponding to a solution u0 = u0 (t; r, z) with r ∈ R1 equals ν I (r) = max{Im νj (r) | n < |j| < j1 }.10 Indeed, by the proposition any variation u (t) of a solution u0 (t) can be written as Φ0 (u0 )∗ (r , z ) + Φ1 (u0 )y and in terms of the prime-variables the equation (5.2) reads as r˙ = 0,

z˙ = ω(r)∗ r ,

y˙ = JB(r)y .

(5.16)

Decomposing y (0) in the basis {ψj } we find that e−at u (t)s → 0 as t grows, if a > ν I (r). If a < ν I (r) and ψj is an eigenvector of JB(r) with the eigenvalue νj such that Im νj = a, then y (t) = e−iνj t ψ−j is the y -component of a solution of (5.16). A norm of this solution grow with t faster than eat . 5.4

Lower-dimensional invariant tori of finite-dimensional systems and Floquet’s theorem

Let O be a domain in the Euclidean space R2N , given the usual symplectic structure. Let H1 , . . . , Hn , 1 ≤ n < N , be a system of commuting hamiltonians, defined and analytic in O. Let T n ⊂ O be a torus, analytically embedded in O, which is invariant for all n Hamiltonian vector fields VHj . The vector fields are assumed to be linearly independent at any point of the torus. Under mild nondegeneracy assumptions on the system of hamiltonians (see [Nek]), the torus T n can be proven to belong to an n-dimensional family of invariant n-tori Trn : Trn , 0 ∈ R Rn ; T n = T0n , T n ⊂ T 2n = r∈R

where T 2n is an analytic 2n-dimensional submanifold of O. Moreover, the symplectic form, restricted to T 2n , is nondegenerate and T 2n admits analytic 10

We recall that the functions νj (r) with |j| ≥ j1 are real valued by the assumption b).

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coordinates (r, z), z ∈ Tn , such that for every j = 1, . . . , n the vector field VHj , restricted to T 2n , takes the form l ωjl (r)∂/∂zl (the functions ωjl (r) all are analytic). Instead of presenting here the nondegeneracy assumptions, we just assume existence of a family of invariant n-tori as above. Then for any r there exist linear combinations K1 , . . . , Kn of the original hamiltonians Hj such that for every j the vector field VKj restricted to the torus Trn equals ∂/∂zj . Accordingly, at any point (r, z) ∈ Trn every vector field VKj defines N − n j Floquet multipliers eiλl (r) , l = 1, . . . , N − n, corresponding to directions, transversal to T 2n . 11 For simplicity we assume that T 2n is a linearly stable invariant set of every vector field VKj (so also of every VHj ). Then all the functions λjl (r) are real. The following result is a version of the Floquet theorem “for multidimensional time”. For a proof see [K3] and [KK]. Proposition 3. Under the given above assumptions, every vector field VHj , linearised about its solutions in T 2n , has a complete system of N − n skeworthogonal Floquet solutions with real exponents νj (r). We note that in the finite-dimensional situation which we discuss now, the item 2) of Definition 3 becomes trivial.

6

Linearised Lax-integrable equations

Now we pass to the problem of constructing complete systems of Floquet solutions of infinite-dimensional sustems. If (3.1) was a finite-dimensional system (i.e., dim Z < ∞) with an integrable subsystem T 2n as above, then by Proposition 2 linearised equation (5.2) would have a complete system of Floquet solutions provided that the equation (3.1) had n nondegenerate integrals in involution. For infinite-dimensional systems the Floquet theorem is unknown. In this section we show that for Lax-integrable equations Floquet solutions can be constructed as quadratic forms of the eigen-functions of the corresponding L-operator. 6.1

Abstract situation

Let u(t) be any smooth solution of a Lax-integrable equation (5.1)=(2.6). % % ∂ For any smooth vector v we denote Lt (v) = Lu(t) (v) = ∂ε Lu(t)+εv % , and ε=0

11

The multipliers are defined as eigenvalues of the linearized time-2π flow-map of the vector field VKj , restricted to a skew-orthogonal component to the space T(r,z) T 2n . They are z-independent, see [K3].


97

similar define operators At (v). Differentiating equation (3.4) in a direction v, we get a Lax-representation for the linearised equation (5.2): d L (v) = [At (v), Lt ] + [At , Lt (v)], dt t where At = Au(t) and Lt = Lu(t) . Let us consider smooth eigenvectors of the operator L0 = Lu0 and of its conjugate operator L∗0 , corresponding to the same eigenvalue λ: L0 χ0 = λχ0 , L∗0 ξ0 = λξ0 . We assume that the following initial-value problems, χ(t) ˙ = At χ(t),

χ(0) = χ0 ,

˙ = −A∗ ξ(t), ξ(t) t

ξ(0) = ξ0 ,

(6.1)

have smooth solutions χ(t) and ξ(t). Then for any t we have Lt χ(t) = λχ(t) and L∗t ξ(t) = λξ(t) (see Lemma 2.3 for the proof of the first relation; proof of the second is identical). We claim that d L (v(t))χ, ξ = 0. (6.2) dt t Indeed, abbreviating Lt (v(t)) to L and At (v(t)) to A , we write the left-hand side of (6.2) as ˙ + L˙ χ, ξ + L χ, L χ, ξ ˙ ξ = L χ, −A∗ ξ + ([A , L] + [A, L ])χ, ξ + L Aχ, ξ = [A , L]χ, ξ = A Lχ, ξ − A χ, L∗ ξ = (λ − λ)A χ, ξ = 0. Since Lt (w) linearly depends on w ∈ Zs as an operator from Zs to Zs −d , then Lt (w)χ, ξZ = w, qt (χ, ξ)Z

∀ w,

(6.3)

where qt (χ, ξ) = qu(t) (χ, ξ) is an Z−s -valued quadratic form of χ, ξ ∈ Zs , which is C 1 -smooth in t. Hence, we can rewrite (6.2) as d v(t), qt (χ, ξ) ≡ 0. dt For a moment let us denote qt (χ, ξ) = w. Then v, At Jw = −JAt v, w = −v, ˙ w = v, w, ˙

(6.4)

(6.5)

where the last equality follows from (6.4). At this point we assume that the flow-maps Sτt ∗ (u(τ )) of the linearised equation (5.2) preserves the space Z∞ . Then the set {v(t)} formed by values at time t of all smooth solutions of equation (5.2) equals Z∞ , so w˙ = A(t)Jw since (6.5) holds for any t and v(·). Therefore J w˙ = JA(t)Jw, i.e. the curve Jw(t) = for all solutions J q(χ(t), ξ(t)) satisfies the equation (5.2). Thus, linearised Lax-integrable equations have solutions which can be obtained as bilinear forms of eigen-functions of the L-operator and its adjoint:

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Theorem 3. If flow-maps of the linearised equation (5.2) preserve the space Z∞ and the curves χ(t), ξ(t) are smooth solutions of equations (6.1), then the function J qt (χ(t), ξ(t)) with qt defined in (6.3) solves the linearised equation (5.2). Remark. Let {Zs } be a Sobolev scale of 2π-periodic functions of a spacevariable x. If the flows of linear equations (5.2) and (6.1) preserve Sobolev spaces of 2π-antiperiodic functions, the curves χ(t), ξ(t) are constructed as above and (6.3) holds with some 2π-periodic function qt (χ, ξ)(x), then due to the same arguments as before the function J(q) solves (5.2). Remarkably the solutions given by the theorem and the remark have the Floquet form (5.3) and jointly form a complete non-degenerate family. Below we check this property for the KdV and SG equations. 6.2

Linearised KdV equation

Now we consider the KdV equation and take for the invariant manifold T 2n a bounded part of any finite-gap manifold TV2n of the form (4.5). We have already checked that it satisfies assumptions i)–iv) from Section 5. For any n-gap solution u0 (t, · ) ∈ TVn (r) the equation linearised about u0 takes the form v˙ =

1 3 ∂ vxxx + (u0 (t, x)v). 4 2 ∂x

(6.6)

Since u0 (t, x) is a smooth function, then this equation is well-defined in Sobolev spaces H0d with d ≥ 1. Thus the assumption v) on the invariant manifold also is satisfied. The equation (6.6) has trivial solutions, obtained by differentiations in directions, tangent to the n-gap tori. They can be written as ∂u(t,x;r,z) (see ∂zj (4.4)). We recall that the L-operator of the KdV equation is the Sturm–Liouville operator L = −∂ 2/∂x2 − u0 (t, x) and consider any its complex eigenfunction χ(x; λ) with an eigenvalue λ, satisfying the Floquet–Bloch boundary conditions: Lχ(x; λ) = λχ(x; λ),

χ(x + 2π; λ) = eiρ χ(x; λ), ρ = ρ(λ).

(6.7)

One of the most important and elegant properties of the KdV equation (and of the whole class of Lax-integrable equations) is that χ as a function of λ is meromorphic in Γ \ ∞ (Γ = Γ(r) is the Riemann surface defined in√Section 4) and can be normalised to have at infinity the singularity exp i λ x. This function admits a representation in terms of the same theta-function θ and the vectors V , W , z as in Section 4. The representation is given by the


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Its-Matveev formula (see in [DMN,D,BB]): χ(t, x; r, z; P ) = eΩ1 (P )x+Ω3 (P )t

θ(A(P ) + i(V x + W t + z))θ(iz) , θ(A(P ) + iz)θ(i(V x + W t + z)) P = (λ, µ) ∈ Γ.

Here A(P ) is the Abel transformation, the same as above, and Ω1 , Ω3 are Abel integrals of the differentials dΩ1 , dΩ3 . The integrals are defined modulo periods of the differentials. For P = (λ, µ) with real λ we normalise the integrals in the following way:

λ

Ωj (λ, µ) =

dΩj

for (λ, µ) ∈ Γ+ , λ ∈ R,

E1

where [E1 , λ] stands for the path in Γ+ through upper edges of the cuts. We denote by σ the holomorphic involution of Γ which transposes the sheets, σ(λ, µ) = (λ, −µ). Denoting for any P = (λ, µ) ∈ Γ by γP the path from σ(P ) to P through E1 , equal to γP = σ(−[E1 , λ]) ∪ [E1 , λ], we get that Ωj (P ) =

1 2

dΩj , γP

since σ ∗ dΩj = −dΩj due to the normalisation (4.2). Now we take a point P close to infinity and denote by µP the path from σ(P ) to P equal to a lift to Γ of the circle in Cλ centred at infinity, which passes through λ and is cut there. The loop γP − µP is contractible in Γ since it envelops all the cuts, so Ωj (P ) = 12 µP dΩj . Using this equality and (4.2) (with c = 0) we get the asymptotics: √ Ω1 (P ) = k + O(k −2 ), Ω3 (P ) = k 3 + O(k −1 ), k = i λ. (6.8) Let us denote f (U ; r, z; P ) =

θ(A(P ) + iU + iz)θ(iz) θ(A(P ) + iz)θ(iU + iz)

(6.9)

and rewrite χ as χ(t, x; r, z; P ) = eΩ1 (P )x+Ω3 (P )t f (V x + W t; r, z; P ).

(6.10)

By the Riemann theorem (see [D,BB]) the first term of the denominator in the right-hand side of (6.9) has exactly n zeroes which form poles of the function P → f and lie in the ovals a1 , . . . , an . Since |θ(iU + iz)| ≥ C(r) > 0 and |θ(A(∞) + iz)| ≥ C(r), then the function f (U ; r, z; P ) is analytic and bounded for r from an appropriate complex neighbourhood of any compact

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subset of R and for (λ, z, U ) from the complex domain { |Im λ|, |Im z|, |Im U | < δ, Re λ > E1 and dist (λ, ∆Vj ) > δ ∀ j} ,

(6.11)

where δ > 0 is sufficiently small. We recall that the closed gaps [λ2j−1 = λ 2j ] are labelled by indices j ∈ NV . So for any P = P±j , where P±j = {± R(λ2j ), λ2j } ∈ Γ, the function χ(t, x; P±j ) must be 4π-periodic in x. Since f is 2π-periodic, we should have Ω1 (P±j ) ∈ 2i Z. This relation holds identically in r. When r tends to zero, λ2j tends to j 2/4 and Ω1 (Pj ) tends to ij/2 (this follows from elementary calculations, see [KK]). Therefore, Ω1 (Pj ) =

i j, 2

j ∈ NV .

(6.12)

Conversely, for any P which meets (6.12) the function (6.10) is 4π-periodic. Since the operator A for the KdV equation is anti selfadjoint, then the second equation in (6.1) coincides with the first and we can take ξ(t) = χ(t). Now the quadratic form q as in Theorem 3 equals χ2 . Finally, since J = ∂/∂x, then the solutions of the linearised equation (5.2)=(6.6) constructed in Theorem 3 are curves vj (t) ∈ Z of the form vj (t, x; r, z) =

(2π)−1/2 2Ω1 (Pj )

∂ e2(Ω1 (Pj )x+Ω3 (Pj )t) f 2 (V x + W t; r, z; Pj ) . ∂x (6.13)

Here j ∈ ZV , Pj = Pj (r) and the first factor in the right-hand side is a convenient normalisation. Thus we have obtained a system of Floquet solutions of the form (5.4),12 where the sections Ψj of the bundle T c H0d |T 2n have the form 2Ω1 (Pj )x e ∂ √ (6.14) Ψj (r, z)(x) = f 2 (V x; r, z; Pj ) , j ∈ ZV , ∂x 2 2π Ω1 (Pj ) P and the exponents νj are νj (r) = −2iΩ3 (Pj ) = −2i E1j dΩ3 . Since the differential i dΩ3 is real (see (4.2 )), then the exponents νj (r) are real for real r and are analytic in r (they have no algebraic singularities). We claim that this system is complete non-resonant. To simplify notation we suppose that V = (1, . . . , n). √ Now the complex basis {ψj | j ∈ Z0 } is the exponential basis ψj = eijx/ 2π. The system of Floquet exponents is non-resonant. To prove the nonresonance we may assume that the vector r is sufficiently small. For any 12

the set of indices ZV which we use now is in obvious 1-1 correspondence with the set Zn .


101

j ∈ ZV = Zn we denote by V (n+1) the (n + 1)-vector (V , j) and view the torus TVn (r) as a degenerate (n + 1)-gap torus TVn+1 (n+1) (r, 0). It can be checked (n+1)

[KK] that the integral which defines νj (r) equals Wn+1 (r, 0). Since the frequency vector ω for finite-gap solutions which fill the torus TVn (r) is ω = W , then the non-resonance relation (5.18) which has to be checked takes the form n

(n+1)

Wl

(n+1)

(r, 0)sl + Wn+1 (r, 0) ≡ 0.

(6.15)

l=1

We can suppose that s = 0; say, s1 = 0. By Lemma 7, for r = (ε, 0, . . . , 0) (n+1) we have: Wl = const + δl,1 8V3 1 ε2 + O(ε4 ). Therefore, the left-hand side of (6.15) equals const + s1 8V3 1 ε2 + O(ε4 ). It does not vanish identically and (6.15) follows. The system is complete. By the Corollary to Lemma 10 we should only check the assumptions 1b) and 2) from Definition 3. The function f ( · ; r, z; P ) converges to unit as r → 0 (as well as the theta-function, see [KK]). Therefore Ψj (x) converges to the complex exponent (2π)−1/2 eijx = ψj (x), so 1b) follows and it remains to check the item 2). Given any γ > 0 we fix a subset R1 R such that mes(R \ R1 ) < γ (see (5.13)). For r ∈ R1 we shall check the properties 2a)–2c). First we show that the map Φ1 defined in (5.8) is close to the embedding ι. Since Ψ−j = Ψj , we have to examine the vectors Ψj with j ∈ NV only. Since the potentials u(x) have zero mean-values, then λ(Pj ) = 14 j 2 + O(j −1 ) √ (see [Ma,MT]) and k(Pj ) = i λ = 2i j + O(j −2 ). Using (6.8) we get that Ω3 (Pj ) = − 8i j 3 + O(j −1 ). So νj (r) = − 14 j 3 + O(j −1 ),

(6.16)

uniformly in r from a complex neighbourhood R1 + δ of R1 . Since the holomorphic differentials dωl have the form dωl = C(λ−3/2 + . . . )dλ (see [S, KK]), then ∞ |A(Pj )| ≤ C1 λ−3/2 dλ ≤ C1 |j|−1 uniformly in r ∈ R1 + δ. (6.17) j 2 /4

Therefore for all P , U , z as in (6.11) and for r from R1 + δ the function f is close to one: |f (U ; r, z; p) − 1| ≤ C|j|−1 .

(6.18)

Using (6.12), (6.18) and the Cauchy estimate we find that the functions Ψj (r, z) defined in (6.14) are close to complex exponents: Ψj (r, z)(x) =

1 ijx e (1 + ζj (r, z)(x)) , 2π

(6.19)

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where |ζj (r, z)(x)| ≤ Cj −1 for r ∈ R1 + δ ⊂ Cn , | Im z| ≤ δ and | Im x| ≤ δ with some j-independent δ and C = C(δ). To check the property 2a) from Definition 3 with ∆ = 1 we shall show that the map aj ζj (x) aj eijx → r 1 is 1-smoothing, i.e., it sends a space H√ H0r+1 (S 1 ). To do it 0 (S ) to the space√ r −1 ijx we observe that in the Hilbert bases {( 2π j ) e }, {( 2π j r+1 )−1 eijx } of the two spaces the map has the matrix M with the entries Mlj = lr+1 j −r ei(j−l)x ζj (x) dx. Since for | Im x| < δ the function ζj is analytic and bounded by Cj −1 , then |mlj | ≤ Cδ (l/j)r+1 e−δ|j−l| . Therefore the l1 norm of any row and any column of the matrix M is bounded by a constant C . Hence, a norm of the map (6.19) as a map from H0r to H0r+1 is bounded by the same C due to the Schur criterion13 and 2a) follows. The property 2b) with κ = 3 easily follows from (6.19). $ = 1 is an immediate consequence The property 2c) with dA +dJ = 3 and ∆ of the asymptotic (6.16).

The system satisfies a)–d). 14 The first assertion of a) follows from the convergence νj (r) = −2iΩ3 (Pj ) → − 14 j 3 as r → 0, which implies that for small r all the functions νj are distinct. The second assertion follows from the item 2c) of Definition 3 which is checked already with dA + dJ = 3 and $ = 1. ∆ The assumption b) follows from 2a), 2c) since the exponents νj are real (for real r). The assumptions c), d) are now empty since all the Floquet exponents are analytic functions. Finally for the domain R as in (4.5) we have proved the following result: Theorem 4. For any γ > 0 there exists a subset R1 R, mes(R \ R1 ) < γ, such that on T12n = Φ0 (R1 × Tn ) ⊂ T 2n the system of Floquet solutions (6.14) with j ∈ ZV is complete non-resonant (in any space H0d , d ≥ 1). 6.3

Higher KdV-equations

The lth equation from the KdV-hierarchy has an [L, A]-pair with the same L-operator L = −∂ 2/∂x2 − u and with some A-operator of the form A = Al = const ∂ 2l+1/∂x2l+1 + . . . (see [DMN,MT,ZM]). Solutions χl of equation (6.1) with A = Al are given by the Its–Matveev formula (6.6), where the differential Ω3 should be replaced by an appropriate differential Ω2l+1 and 13

14

The criterion states that a norm of a linear operator which in Hilbert bases of the two Hilbert spaces has a matrix {Mlj }, is no bigger than (supl j |Mlj | · supj l |Mlj |)1/2 . See [HS] or [KK]. see (5.4) and below.


103

the vector W – by the vector W l , formed by b-periods of Ω2l+1 . We get Floquet solutions vjl of the linearised lth equation, l

vjl (t, x; r, z) = eiνj (r)t Ψj (r, z0 + W l t)(x) ,

j ∈ ZV ,

where νjl = 2Ω2l+1 (Pj ) and Ψj is given by (6.14). Similar to the KdV-case, we find that νjl = 2(i/2)2p+1 j 2l+1 + O(j 2l−3 ),

j ∈ NV .

(6.20)

The system of Floquet solutions {vjl } is complete. Indeed, the items of Definition 3 from 1) through 2b) describe properties of the sections Ψj which are the same as for the KdV equation, so we have already checked them. The $ = −∆ $ l = 2l − 3 follows from (6.20). property 2c) with −∆ The linearised lth equation satisfies the assumption v): its flow-maps Sτt ∗ are well-defined linear isomorphisms. Indeed, by Lemma 8, outside the singular set Rs × Tn the vectors {Ψj (r, z)} form an equivalent complex basis of the skew-normal space Tu⊥c T 2n ⊂ Zd , where d ≥ 1 and u = Φ0 (r, z). After we choose these bases in the spaces Tu⊥c T 2n and Tu⊥c T 2n , the 0 (τ ) 0 (t) l

map Sτt ∗ becomes diagonal with the unit diagonal elements {eiνj (r)(t−τ ) }. So Sτt ∗ (u0 (τ ))d,d ≤ C(r, d) for all τ and t, if d ≥ 1 and r ∈ / Rs . 6.4

Linearised SG equation

Let us consider the SG equation, linearised about any finite-gap solution u(t, x) as in (4.7), (4.9). Now it is more convenient to study the SG equation in the form (3.3). Accordingly, the linearised equation takes the form: ¨˜ = u u ˜xx − (cos u(t, x))˜ u,

˜˙ . w ˜ = −A1/2 u

(6.21)

As in the KdV-case, solutions of equations (6.1), i.e. of the nonautonomous Aequation and its adjoint, are given by explicit theta-formulas (see [EFM, BT]). Accordingly, Theorem 3 provides us with explicit formulas for solutions of equation (6.21) which satisfy the periodic/antiperiodic boundary conditions. Similar to the KdV-case, these solutions have the Floquet form: ˜ u ˜j (t, x; D, r) = G(V˜ x + W t + D; Pj )e(Ω1 +Ω2 )(Pj )W +(Ω1 −Ω2 )(Pj )t , ˜˙ j ; j ∈ Zn . w ˜j = −A−1/2 u (6.22)

Here G(q; p) is some analytic function of q ∈ Tn and P ∈ Γ, expressed via the θ-function, and Ω1 (P ), Ω2 (P ) are Abel integrals on Γ. The points Pj are defined as solutions for the following equation: 1 (Ω1 + Ω2 )(Pj ) = j, i

j ∈ ZV ,

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(cf. (6.12) and [EFM]). Similar to the KdV-case, the Floquet exponents νj (r) = 1i (Ω1 − Ω2 )(Pj ) can be described as the last components of frequency ˜ ∈ Rn+1 corresponding to surfaces T 2n+2 , where except the previvectors W ¯1 ], [E −1 , E ¯ −1 ], . . . , [E −1 , E ¯ −1 ] we also ε-open closed ously opened gaps [E1 , E n n 1 1 gaps at the points Pj and 1/Pj , and next sent ε to zero. In striking difference with the KdV case, some exponents νj are non-real if the open gap are sufficiently large; corresponding functions r → νj (r) have algebraic singularities, see [McK, EFM]. Asymptotical analysis of solutions (6.22) show that after multiplying them by proper constants the solutions take the form u ˜j =

1 i(jx+νj (r)t) e (1 + ζj (r, D)(x)) , w ˜j = . . . , 2π

(6.23)

where the function ζj goes to zero when j → ∞ or r → 0. Due to the same arguments as in the KdV case, the system of Floquet solutions (6.23) satisfies the asymptotic assumption 2) from Definition 3, and due to (4.10) it is nonresonant (see [BiK1, BoK2, KK]). Using the Corollary from Lemma 10, we get that this system of Floquet solutions of the linearised equation (SG)+(OP) is complete nonresonant.

7 7.1

Normal form A normal form theorem

We continue to study the Hamiltonian equation (5.1) near an invariant manifold T 2n = Φ0 (R × Tn ) which possesses the properties (i)–(v) as in Section 3. Proposition 2 puts the linearised equation (5.2) to a constant coefficient normal form, provided that this equation possesses a complete non-resonant system of Floquet solutions. In this section we show that under these assumptions the equation (5.1) itself can be put to a convenient normal form in a neighbourhood of T 2n . Namely, we show that the action-angle variables (p, q) on T 2n can be supplemented by a skew-orthogonal to T 2n vector-coordinate y in such a way that in the new coordinates the equation is Hamiltonian with a hamiltonian h(p) + 12 B(p)y, y + h3 (p, q, y),

h3 = O(y3 ) .

Here B(p) is the self-adjoint operator from Proposition 2 and the term h3 defines a hamiltonian vector field of the same order as the nonlinear part J∇H of the original equation (this is a crucial property of the normal form!). The normal form is an effective technical tool to study equation (5.1) and its perturbations in the vicinity of T 2n . We assume that the linearised equation (5.2) has a complete family of skew-orthogonal Floquet solutions vj (t) as in (5.4), define the singular subset Rs , Rs = Rsc ∩ R as in Section 5 (see there a remark after Definition 3). As


105

in Section 5, we choose any domain R1 in a compact part of the regular set R0 = R \ R s . By Lemma 5 the equation (3.1) is integrable in Φ0 (R0 × Tn ). So we can cover Φ0 (R1 × Tn ) by a finite system of open sub-domains such that in each one the equation admits analytic action-angle variables (p, q). For simplicity we suppose that the action-angles exist globally in Φ0 (R1 ×Tn ) and use them instead of (r, z). Accordingly, we write Φ0 (R1 × Tn ) as Φ0 (P × Tn ), where P = {p} Rn and Tn = {q}. We denote W = P × Tn and Φ0 (W ) = T12n . The map Φ0 W → T 2n ⊂ Z analytically extends to a bounded analytic map W c → Z c , where W c is a complex neighbourhood of W , W c = (P + δ) × {q ∈ Cn/2πZn | | Im q| < δ}. Since ω = ∇h (see Lemma 5), then we write the skew-orthogonal Floquet solutions vj (t) as vj (t; p, q) = eiνj (p)t Ψj (p, q + t∇h(p)) ,

p ∈ P, q ∈ Tn , j ∈ Zn .

(7.1)

The linear in y map y → Φ1 (p, q + t∇h)y as in (5.8) reduces the linearised equation (5.2) to the constant-coefficient linear equation y˙ = JB(p)y, . . . ,

(7.2)

where the dots stand for components of the linearised equation in directions tangent to W × {0} (see Proposition 2). We denote by Sδ = Sδ (Yd ) the manifold Sδ = W × Oδ (Y ), Y = Yd , and denote by Sδc its complex neighbourhood Sδc = W c × Oδ (Y c ). We give Sδ symplectic structure by means of the 2-form (dp∧dq)⊕α2Y , where α2Y = α2 |Y . Our goal in this section is to prove the following Normal Form Theorem: Theorem 5. Let the Hamiltonian equation (5.1) and its invariant submanifold T 2n satisfy the assumptions i)–v); let a sub-domain T12n = Φ0 (P ×Tn ) ⊂ T02n be as above and (7.1) be a complete system of skew-orthogonal Floquet solutions of the linearised equation (5.2). Then there exists δ1 > 0 and an analytic symplectomorphism G (Sδ1 , dp∧dq ⊕α2Y ) → (Z, α2 ) such that G(Sδ1 ) is a neighbourhood of T12n and H ◦ G = h(p) + 12 B(p)y, y + h3 (p, q, y). Here h3 = O(y3 ) is an analytic functional such that its gradient map is of $ − dJ }, i.e. ∇y h3 (p, q, y) ˜ ≤ Cy2 for order d˜ = max{dH , −∆ − dJ , −∆ s s−d any (p, q, y) ∈ Sδ1 . To simplify our presentation we suppose below that all the frequencies νj (p) are real and consequently the operator B(p) is diagonal in the ϕj -basis ν (p) of the space Y , i.e., B(p)ϕj = jν J ϕj for every j. j

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We start with the affine in y ∈ Y map Φ, Φ = Φ0 + Φ1 Sδc → Z c ,

(p, q, y) → Φ0 (p, q) + Φ1 (p, q)y.

It is real (it sends Sδ to Z), bounded on bounded subsets of Sδc and is weakly analytic by assumptions b) and d). So Φ is an analytic map by the criterion of analyticity. By Lemma 8 its linearizations at points from W × {0} define isomorphisms of R2n × Y and Z. Thus, by the inverse function theorem the map Φ defines an analytic isomorphism of Sδc and a complex neighbourhood of T12n in Z, provided that δ is sufficiently small Next we study symplectic properties of the map Φ. Since restriction of Φ to W × {0} equals Φ0 and restriction to any disc {w} × Oδ (Y ) equals Φ1 (w) up to a translation, then these restrictions are symplectic. In particular, for any w ∈ W the map Φ∗ (w, 0) is a linear symplectomorphism. Hence, the pull-back form ω2 := Φ∗ α2 equals (dp ∧ dq) ⊕ α2Y for w = 0 and these two forms coincide being restricted to any disc {w} × Oδ (Y ). It means that the difference ω∆ = ω2 − dp ∧ dq ⊕ α2Y may be written as ω∆ = jW W (w, y)dw ∧ dw + jW Y (w, y)dy ∧ dw + jY W (w, y)dw ∧ dy, where the linear operators jW W (w, y), jW Y (w, y) and jY W (w, y) vanish for y = 0. In the calculations we carry out below it is convenient to adopt gradientnotations for linearizations of the maps Φ and Φ1 in w. Namely, below we write Φ(w, y)∗ (δw, 0) = ∇wj Φ(w, y)δwj =: ∇w Φ · δw. Here ∇w Φ = (∇p Φ, ∇q Φ) ∈ Z × · · · × Z (2n times). Similar we write Φ1∗ (δw, 0) = ∇w Φ1 · δw, where any ∇wj Φ1 is a linear operator Y → Z. In these notations we have: ¯ 1 δy, ∇w Φ1 y · δw ω2 [δy, δw] = α2 [Φ1 δy, Φ0∗ δw + (∇w Φ1 · δw)y] = JΦ ¯ w Φ1 · δw)y, Φ1 δy. Hence, and ω2 [δw, δy] = J(∇ jW Y (w, y)δy = JΦ1 (w)δy, ∇w Φ1 (w)y, jY W (w, y)δw = Φ∗1 (w)J(∇w Φ1 · δw)y.

(7.3)

Abbreviating (δw, δy) ∈ R2n × Y to δz, we write the form ω∆ as ω∆ = J ∆ dz ∧ dz, where J ∆ is the operator matrix: ( ' jW W jW Y . (7.4) J ∆ = J ∆ (w, y) = ∗ −jW 0 Y The form ω∆ is exact, as well as the forms α2 and dp∧dq ⊕w2Y , i.e. ω∆ = dω1 . Lemma 2 represents the 1-form ω1 as 1 ) * ω1 (w, y) = JΦ1 (w)y, ∇w Φ1 (w)ty dt dw =

0 1 2 JΦ1 (w)y, ∇w Φ1 (w)ydw

= 12 α2 [Φ1 (w)y, ∇w Φ1 (w)y]dw.


107

We have seen that ω2 = Φ∗ α2 = (dp ∧ dq) ⊕ α2Y + d(L(w, y)dw),

(7.5)

where the 2n-vector L has the components Lj = 12 α2 [Φ1 (w)y, ∇wj Φ1 (w)y]. Next we calculate how the map Φ changes the hamiltonian H. To begin with we analyse how Φ1 transforms the quadratic part Au, u of the hamiltonian H. Since the nonautonomous symplectic linear map Φt := Φ1 (p, q + t∇h) sends solutions y(t) of equation (7.2) to solutions v(t) = Φt y(t) of (5.2), then we have the equalities: v˙ + + +

Φ˙ t y + Φt y˙ + + +

JAt v + + +

Φ˙ t y + Φt JB(p)y

JAt Φt y Thus, JAt Φt y = Φ˙ t y + Φt JB(p)y.

(7.6)

Taking skew-product of (7.6) with −v, we get: JAt Φt y, Jv + + + ∗

Φt At Φt y, y

Φ˙ t y + Φt JB(p)y, Jv + + +

(7.7)

Φ˙ t y, JΦt y + B(p)y, y,

¯ t y = JBy, Jy ¯ = By, y by symplecticity of where we use that Φt JBy, JΦ t the map Φ . Since for t = 0 we have At = A + ∇H(Φ0 (w))∗ and Φ˙ t = ∇q Φ1 (w) · ∇h(p), then relation (7.6) with t = 0 implies that Φ1 (w)JB(p) = J(A + ∇H(Φ0 (w))∗ )Φ1 (w) − ∇q Φ1 (w) · ∇h(p). Similar, (7.7) implies that ) * B(p) − Φ1 (w)∗ (A + ∇H(Φ0 (w))∗ )Φ1 (w) y, y = Φ1 (w)∗ J ∇q Φ1 (w) · ∇h(p) y, y = A(w)y, y, ¯ q Φ1 · ∇h), i.e., where A stands for the symmetrisation of the operator Φ∗1 J(∇ ¯ q Φ1 (w) · ∇h(p)) − (∇q Φ1 (w)∗ · ∇h(p))JΦ ¯ 1 (w) . A(w) = 12 Φ∗1 (w)J(∇ Since this relation holds identically in y ∈ Y , then B(p) − Φ1 (w)∗ (A + ∇H(Φ0 (w))∗ )Φ1 (w) = A(w).

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Lemma 11. The operator A defines a (∆ + dJ )-smoothing symmetric map c A : Ydc → Yd+d , analytic in w ∈ W c . J +∆ Proof. The operator A is symmetric by its construction. It remains to check its smoothness. Since ∇q Φ1 = ∇q (Φ1 − ι), then by (5.10) and the Cauchy estimate the operator ∇q Φ1 · ∇h analytically depends on w ∈ W c as a map Ydc → Zd+∆ . c c By Lemma 9 the operator Φ1 (w)∗ J¯ : Zd+∆ → Yd+d also is analytic in w. J +∆ Hence, the first term of the operator A defines an analytic in w ∈ W c map c Ydc → Yd+d . J +∆ ¯ 1 (w) : Y c → Z c Using Lemma 9 once again we find that the operator JΦ d d+dJ is analytic in w. Due to the second assertion of this lemma and the Cauchy c c estimate, the map ∇q Φ∗1 · h : Zd+d → Yd+d is analytic in w as well. J J +∆ Combining these two statements, we find that the second term of A also c defines an analytic in w ∈ W c map Ydc → Yd+d . This completes the J +∆ proof. Now we write the transformed hamiltonian as H ◦ Φ = 12 AΦ0 , Φ0 + AΦ0 , Φ1 y + 12 AΦ1 y, Φ1 y + H(Φ), and separate its affine in y part: H◦Φ=

1 2 AΦ0 , Φ0

+

! ! + H(Φ0 ) + AΦ0 , Φ1 y + ∇H(Φ0 ), Φ1 y

1 2 AΦ1 y, Φ1 y

! + H(Φ0 + Φ1 y) − H(Φ0 ) − ∇H(Φ0 ), Φ1 y .

The first term in the right-hand side equals h(p). By Lemma 2 the form ω2 = Φ∗ α2 equals (dp ∧ dq) ⊕ α2Y , when y = 0. Hence, for y = 0 the y-component of equation (5.1), written in the (p, q, y)variables, is J∇y (H ◦ Φ). It equals zero since the set {y = 0} is invariant for the equation. Thus, the second term vanishes. By Lemma 11, AΦ1 y, Φ1 y = By, y − ∇H∗ Φ1 y, Φ1 y − Ay, y . Therefore the third term in the r.h.s. equals 12 B(p)y, y + h2 (p, q, y), where h2 = − 12 ∇H(Φ0 )∗ Φ1 y, Φ1 y − 12 A(w)y, y + H(Φ0 + Φ1 y) − H(Φ0 ) − ∇H(Φ0 ), Φ1 y. It is easy to see, using Lemmas 9 and 11, that h2 defines an analytic gradient map ∇y h2 R2n × Yd → Yd−d˜. Thus, the affine in y map Φ transforms the hamiltonian H to H ◦ Φ = h(p) + 12 B(p)y, y + h2 (p, q, y), ˜ where h2 = Oy2 and ord ∇h2 = d. Our next goal is to normalise the symplectic structure ω2 = Φ∗ α2 in Sδ by means of the Moser–Weinstein theorem (Lemma 3). The theorem states that


109

ϕ∗ ω2 = (dp ∧ dq) ⊕ α2Y , where ϕ is the time-one shift S01 along trajectories of a nonautonomous equation: z˙ = V t (z) , z = (w, y). The vector field V t Sδ → R2n × Yd is obtained as a solution of the equation −(J 0 + tJ ∆ )V t = a(z, y), where J 0 (δp, δq, δy) = (−δq, δp, Jδy), the operator J ∆ is as in (7.4) and the map a is such that differential of the 1-form a(z)dz equals ω2 − (dp ∧ dq) ⊕ α2Y . By (7.5), a(z) = (L(z), 0). We claim that the map ϕ sends Sδc1 to Sδc (δ1 is sufficiently small compare to δ) and transforms H ◦ Φ to a hamiltonian of similar form: Lemma 12. The hamiltonian H ◦ Φ ◦ ϕ equals to H ◦ Φ ◦ ϕ = h(p) + 12 B(p)y, y + 12 B(p, q)y, y + h3 (p, q, y),

(7.8)

where B(p) is the same as in (7.2) and B(p, q) is a linear operator of or˜ analytic in (p, q) (d˜ is the same as in Theorem 5)). The function der d, ˜ ∇y h3 (p, q, y) ˜ ≤ h3 = O(y3 ) has an analytic gradient map of order d, d+d 2 Cy . The statement of the lemma is quite obvious for a finite-dimensional phase space Sδ , but not in the infinite-dimensional situation. Indeed, the transformation ϕ has the form ϕ = id +ϕ, $ where ϕ $ = y2 is a ∆-smoothing map. Thus the transformed hamiltonian gets the term B(p)y, ϕy $ which is O(y3 ) with the gradient of order dA − ∆. The number dA − ∆ could be rather big and the term could spoil the forthcoming constructions. Fortunately, it vanishes up to a smoother term. This is essentially what the lemma states. For its proof see Lemma 7 in [K1] and [KK]. To prove the theorem it remains to check that the operator B in (7.8) vanishes. Since ϕ is analytic and O(y2 )-close to the identity, then ϕ∗ (w, 0)|{0}×Y = id . Thus the transformation y(t) → (Φ ◦ ϕ)∗ (p, q + t∇h, 0) y(t) sends solutions of the equation (7.2) to solutions of (5.2). From other side, Φ ◦ ϕ is a canonical transformation which transforms solutions of the equation with hamiltonian (7.8) to solutions of (5.1). In particular, it sends the curves w(t;p, q) = (p, q + t∇h(p), 0) to solutions u0 (t) of (5.1). Hence, the linearisation Φ ◦ ϕ(w(t)) ∗ transforms solutions of the linearised equation y˙ = J B(p) + B(w(t) y, . . . (7.9) to solutions of (5.2). Therefore the transformation ϕ(w(t))∗ sends solutions of (7.9) to solutions of (7.2). Since a y-component of the map ϕ(w(t))∗ is the identity, then we must have JB(p)y ≡ J(B(p) + B(p, q + t∇h))y. This implies that B ≡ 0 and the theorem is proven.

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S. B. Kuksin

Examples

1)Korteweg–de Vries equation. The KdV equation in a Sobolev space Zs = H0s (S 1 ) with s ≥ 3, which is given symplectic structure by the form α2 = (−∂/∂x)−1 du, duL2 takes the form (3.1) (see Example 3.1). Its restriction to a bounded part T 2n of any finite-gap manifold TV2n (see (4.5)) satisfies the restrictions i)–v) and the corresponding system of Floquet solutions (6.13) $ = 1, dJ = 1, dH = 0 and dA = 2. is complete non-resonant with ∆ = ∆ Therefore Theorem 5 provides KdV with a normal form. To state the result, we find the singular subset Rs ⊂ R 15 and choose any domain R1 R \ Rs . We cover R1 up to its zero-measure subset by non-overlapping sub-domains R11 , R12 , . . . such that the KdV-equation restricted to any manifold Φ0 (R1j × Tn ) = Jj2n admits action-angle variables (p, q) with p ∈ Pj Rn . For any s we denote by Ys ⊂ H0s (S 1 ) the closed subspace spanned by the functions {cos jx, − sin jx | j ∈ NV }. Applying Theorem 5 we get: Theorem 11. For any s ≥ 3 there exists δ > 0 and an analytic symplectomorphism G of (Pj ×Tn ×{ys < δ}, dp∧dq⊕α2Y ) and a domain in (H0s , α2 ). It contains Tj2n in its range and G−1 transforms KdV to the Hamiltonian system q˙ = ∇p H,

p˙ = −∇q H,

y˙ =

∂ ∇y H ∂x

(7.10)

with a hamiltonian H of the form H = h(p) + 12 B(p)y, y + H3 (p, q, y). Here h(p) is the hamiltonian of KdV restricted to Tj2n , B(p) is the linear operator in Yd with eigenvectors cos mx, − sin mx and eigenvalues νm (p) (m ∈ NV ), H3 = O(y3s ) is a function with a zero-order analytic gradient map. 2) Higher KdV equations. Let us take any lth equation from the KdVhierarchy. Since the same (as in the KdV-case) sections Ψj of the skew-normal bundle to a finite-gap manifold TV2n give rise to its Floquet solutions, then the same map Φ1 reduces the linearised lth equation to the equation y˙ = JB l (p)y in the space Y . Here J = ∂/∂x and B l (p) is a linear operator with the eigenvectors cos jx, − sin jx, corresponding to the eigenvalues νjl (p) (j ∈ NV ) as in (6.20) Therefore, the same map G with s ≥ 2p + 1 reduces the lth KdV equation in the vicinity of Tj2n (the same as above part of TV2n ) to the equation (7.10) with H = Hl (p, q, y) = hl (p) + 12 B l (p)y, y + H3l (p, q, y). Here hl is the hamiltonian of the lth equation restricted to Tj2n and the operator B l (p) has the eigenvalues νjl . Now ∆ = dJ = 1 as in the KdV-case, dA = 2l, $ = 3 − 2l by (6.20). So d˜ = 2l − 2 and H3 = O(y3 ) has dH = 2l − 2 and ∆ s an analytic gradient map of order 2l − 2. 3) Sine-Gordon equation. 15

In the KdV-case the set Rs is empty. We neglect this nice specificity of KdV since our goal is to present arguments applicable to other integrable PDEs.


8 8.1

111

The KAM theorem The main theorem and related results

Let ({Zs }, α2 ), α2 = J dz ∧ dz be a scale of symplectic Hilbert spaces and ord J¯ = −dJ ≤ 0. Let H be a quasilinear hamiltonian of the form H = 1 2 Az, z + H(z), where A is a selfadjoint isomorphism of the scale of order dA > −dJ . We fix any d ≥ dA /2 and assume that the function H is analytic in the space Zd (or in a neighbourhood in Zd of the manifold T02n , see below) and defines an analytic gradient map of order dH , ∇H : Zd → Zd−dH . We have dH < dA due to the quasilinearity of the hamiltonian H. The corresponding Hamiltonian equation takes the form: u˙ = J∇H(u) = J(Au + ∇H(u)),

J = (−J)−1 .

(8.1)

As in Sections 3 and 5 we assume that the equation (8.1) has an invariant manifold T02n = Φ0 (R0 × Tn ) filled with quasiperiodic solutions u0 (t; r, z) which satisfies the assumptions i)–v). The manifold R0 is the regular part $ we denote any chart on R0 of an n-dimensional real analytic set R. By R analytically diffeomorphic to a bounded connected subdomain of Rn . We $ with this domain and supply it with the n-dimensional Lebesgue identify R measure mesn . As in Section 5, we also consider linearisation of the equation (8.1) about a solution u0 : (8.2) v˙ = J Av + ∇H(u0 (t))∗ v , and assume that (8.2) has a system (5.4) of Floquet solutions vj (t). Our concern in this section is a hamiltonian perturbation of the equation (8.1): u˙ = J(Au + ∇H(u) + ε∇H1 (u)),

(8.3)

and behaviour of solution for (8.3) near the manifold T02n . We assume that H1 is an analytic functional such that ord ∇H1 = dH . By d˜ we denote the real number from Theorem 5, d˜ = max {dH , −∆ − $ dJ , −∆−d J }, where −∆ is the order of the linear operator Φ1 −ι (see (5.10)) $ is the exponent of growth in j of “variable parts” of the the Floquet and ∆ exponents νj (r) (see (5.12)). Let us fix any ρ˜ such that 0 < ρ˜ < 1/3. Now we state a KAM theorem which is the main result of this paper: Theorem 7 (the Main Theorem). Let the invariant manifold T02n satisfy the assumptions i)–v) and the system of Floquet solutions (5.4) for (8.3) is complete nonresonant. Besides, d1 := dA + dJ ≥ 1 and 1 2 1) (spectral asymptotic): νj (r) = K1 j d1 + K11 j d1 + K12 j d1 + · · · + ν˜j (r), where K1 > 0, d1 > d11 > . . . (the dots stand for a finite sum), the functions ν˜j analytically extend to Rc , where |˜ νj | ≤ Cj κ with some κ < d1 − 1;

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S. B. Kuksin

2) (quasilinearity): d˜ < d1 − 1. Then most of the invariant tori T n (r) of equation (8.1) persist in (8.3) $ ⊂ R0 as above and any when ε → 0 in the following sense: for any chart R $ε R $ and a Lipschitz embedding sufficiently small ε > 0, a Borel subset R $ε × Tn −→ Zd can be found such that: Σε : R $\R $ε ) −→ 0 as ε −→ 0, a) mesn (R $ε × Tn −→ Zd is bounded by Cερ˜ as well as its b) the map (Σε − Φ0 ) : R Lipschitz constant and is analytic in q ∈ Tn ; $ε , is invariant for the equac) each torus Tεn (r) := Σε ({r} × Tn ), r ∈ R tion (8.3) and is filled with its time-quasiperiodic solutions hε (t) of the form hε (t) = hε (t; r, z) = Σε (r, z + tωε (r)), where |ωε − ω| + Lip (ωε − ω) ≤ Cερ˜. Amplification. The statements b), c) of Theorem 7 remain true with ρ˜ replaced by any ρ < 1. Besides, Σε − Φ0 d + |ωε − ω| ≤ Cε. n , ), W , = R×T $ ,ε ), W ,ε = R $ε ×Tn . and T$ε2n = Σε (W We denote T$ 2n = Φ0 (W 2n 2n The set T$ε is a remnant of the invariant manifold T$ in the perturbed equation (8.3). Since T$ 2n is a 2n-dimensional manifold embedded to Zd , then its 2n$ 2n is finite and positive. The remnant dimensional Hausdorff measure mesH 2n T 2n $ set Tε is very irregular (it is totally disconnected). Still it carries most of a measure of the set T$ 2n :

$ 2n is bigger Proposition 4. Under the assumptions of Theorem 7, mesH 2n Tε than mesH T$ 2n − o(1) as ε → 0. 2n

Proof.

16

By the assertion a) of the theorem, , , mesH 2n (W \ Wε ) = o(1).

(8.4)

,ε →−→ Φ0 (W ,ε ) ⊂ T$ 2n is Lipschitz and has a LipsThe map Φ0 : W ∼

,ε ) →−→ T$ 2n has the form : Φ0 (W chitz inverse, so the map Σε ◦ Φ−1 ε 0 ∼

$ 2n ≥ id + L, where Lip L ≤ Cερ˜ (we use the assertion b)). Therefore, mesH 2n Tε H ρ ˜ H ,ε ) − O(ε ). Since mes (Φ0 (W \ W ,ε )) = o(1) by (8.4), then the mes2n Φ0 (W 2n assertion follows. Under the assumptions of Theorem 7, a solution u0 (t; r, z) of (8.1) is linearly stable if all Floquet exponents νj (r) are real (see the Corollary to Propo$ Then the solutions sition 2). Let us assume that this is the case for all r ∈ R. $ε also are linearly stahε (t; r, z) of the perturbed equation (8.3) with r ∈ R ble, provided that this equation linearised about hε satisfies some a priori estimate. We recall that by the assumption v) the flow maps Sτt ∗ (hε (τ )) of 16

For basic properties of the Hausdorff measure which we use below see e.g. [Fal].


113

the linearised equation are well defined in the space Zd . We say that the linearised equation is uniformly well defined (in Zd ) if Sτt ∗ (hε (τ ))d,d ≤ C1 eC2 t

for all t, τ.

(8.5)

Theorem 8. If under the assumptions of Theorem 7 all the Floquet expo$ then a solution hε (t) is linearly stable, nents νj (r) are real for all r ∈ R, provided that equation (8.3) linearised about this solution is uniformly well defined. (Examples we consider below in Section 9 show that the assumption (8.5) is quite non-restrictive). We prove the two theorems and the amplification, reducing them to similar statements concerning perturbations of parameter-depending linear systems. 8.2

Reduction to a parameter-depending case

We perform the reduction in four steps. Step 1 (localisation). Let us denote by Rf the set of singularities of the $ | det ω∗ (r) = 0}, and denote R $s = (Rs ∩ R) $ ∪ frequency map ω, Rf = {r ∈ R Rf , where Rs is the singular set, constructed in Section 5. By the assumption $ So R $s also is one, and for any iv), Rf is a proper analytic subset of R. given positive γ0 we can find a finite system of M connected subdomains $l ⊂ R $\R $s such that dist (rj , rj ) ≥ C(γ0 ) > 0 if rj ∈ R $j and rj ∈ R $j with R j = j. Besides, ˜ \ ∪R $l ) < γ0 , a) mes (R $l ×Tn ) admits analytic actionb) the hamiltonian system restricted to Φ0 (R n angle variables (p, q), where p ∈ Pl R and q ∈ Tn . The map (p, q) → (r, z) has the form r = r(p), z = q+z0 (p). This map, its inverse and the hamiltonian h = hl (p) all are δ-analytic with some positive δ = δ(γ0 ). By Lemma 2.2, ∇h(p) ≡ ω(r(p)); c) for every l the gradient map p → ∇h(p) ≡ ω(r(p)) defines a diffeomorphism Pl →−→ Ωl Rn which is δ-analytic as well as its inverse; ∼

$l is connected, then the eigen-vectors ψj of the d) since each domain R $l . operator JB(r) are r-independent when r ∈ R Step 2 (a normal form theorem). In this step of the proof and in the next $l as above and drop the index l. Step 3 we consider any fixed domain R Applying Theorem 5 we find an analytic symplectomorphism G which $ × Tn ) to the form transforms the equation (8.1) in the vicinity of Φ0 (R given in the theorem. The same symplectomorphism converts the perturbed equation (8.3) to the Hamiltonian system p˙ = −∇q Hε , q˙ = ∇p Hε , y˙ = J∇y Hε .

(8.6)

Here p ∈ P , q ∈ Tn , y ∈ Oδ (Yd ) and Hε = h(p) + 12 B(p)y, y + h3 (p, q, y) + ˜ The operator B(p), the εH1 (p, q, y) with h3 = O(y3d ) and ord ∇y h3 = d.

114

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functions h, h3 , H1 and their gradients all are δ-analytic in the corresponding domains. Step 3 (introducing a parameter). Let us consider the following neighbourhoods of the torus T0n = {0} × Tn × {0} in Y and Y c : Qδ = Oδ (Rn ) × Tn × Oδ (Yd ) ⊂ Y, Qcδ = Oδ (Cn ) × {|Im q| < δ} × Oδ (Ydc ) ⊂ Y c . In the equation (8.6) we perform a shift of the action p: p, q˜, y˜), (p, q, y) = (˜ p + a, q˜, y˜) =: Shifta (˜ where a ∈ P is a parameter of the shift. After this transformation hamiltonian p, q˜, y˜; a) of the tilde-variables from the Hε becomes an analytic function Hε (˜ domain Qcδ . It has the following form: ˜ 3 (˜ y , y˜ + εH1 (˜ p + a, q˜, y˜) + h p, q˜, y˜; a), Hε = h(a) + ∇h(a) · p˜ + 12 B(a)˜ ˜ 3 ˜ = O(˜ ˜ 3 = O(˜ y 3d + |˜ p|2 + |˜ p|˜ y 2d ) and ∇y h y 2d + |˜ p|˜ y d ) (so where h d−d ˜ ˜ ord ∇h3 = d). ˜ 3 and the Floquet exponents νj are analytic bounded The functions h, H1 , h functions of the parameter a ∈ P +δ. Because the property c) from Step 1, the map P a → ω = ∇h(a) ∈ Ω defines an analytic Lipschitz diffeomorphism of P and a bounded domain Ω ⊂ R. We drop the tildes and change the parameter a to ω. Now the hamiltonian Hε reeds as Hε (p, q, y; ω) = h(a) + ω · p + 12 B(ω)y, y + εH1 (p, q, y; ω) + h3 (p, q, y; ω). The operator JB is diagonal in the symplectic basis {ψj }, constructed in Proposition 2: Jψj = ±iνjJ ψj , JBψj = νj (r)ψj

∀ j ∈ Zn .

Since the hamiltonian Hε is δ-analytic, then by the Cauchy estimate it is Lipschitz in a ∈ P as well as in ω ∈ Ω. This is all we need from its dependence in the parameters. In the vicinity of the torus T0n = {0} × Tn × {0} in Qδ the hamiltonian Hε is a perturbation of the q-independent hamiltonian H0 = ω · p + 12 B(ω)y, y (modulo the irrelevant constant h(a)). Indeed, ε is small and the term h3 has on T0n a high-order zero. The hamiltonian equations with the hamiltonian Hε (p, q, y; ω) take the form: p˙ = −∇q (εH1 +h3 ), q˙ = ω + ∇p (εH1 + h3 ), y˙ = J(B(ω)y + ε∇y H1 + ∇y h3 ).

(8.7)


115

We abbreviate (p, q, y) to h and rewrite (8.7) as h˙ = VHε (h). In the context of equations (8.7), we call the functions νj (ω) frequencies of the linear equation. Hamiltonian vector fields with hamiltonians of the form Hε are studied in [K, K2, P]. Now we break the proof of Theorem 7 to present the main result of these works. After this we make the last step to complete the proof of Theorem 7. 8.3

A KAM-theorem for parameter-depending equations

To state the theorem we need, we relax restrictions on the hamiltonian Hε as in the assumptions 1)–3) below: 1) (frequencies). The complex functions νj (ω), j ∈ Zn , are Lipschitz, are real for |j| ≥ j1 with some j1 ≥ n + 1 and are odd in j, νj ≡ −ν−j . For j ≥ n + 1 and for some fixed ω0 ∈ Ω the following estimates hold: 1

2

˜

˜

|νj (ω0 ) − K1 j d1 − K11 j d1 − K12 j d1 − . . . | ≤ Kj d and Lip νj ≤ Kj d , where K1 > 0, d1 ≥ 1, d˜ < d1 − 1 and the dots stand for a finite sum with some exponents d1 > d11 > d21 > . . . . 2) (perturbation). The functions h3 and H1 are analytic in (p, q, y) ∈ Qcδ and everywhere in Qcδ satisfy the estimates:   |H1 | + ∇y H1 d−d+d  ˜ J ≤ 1 ∀ω,    2 2 3  |h3 | ≤ K(|p| + |p|yd + yd ) ∀ω,   2 (8.8) ∇y h3 d−d+d ˜ J ≤ K(|p|yd + yd ) ∀ω,     the same estimates hold for Lipschitz constants    in ω ∈ Ω of these functions and their gradients.  3) (domain of parameters). Ω is a bounded Borel set in Rn of positive Lebesgue measure, such that diam Ω ≤ K2 and |ω| ≤ K for every ω ∈ Ω. Let us choose any ρ ∈ (0, 13 ). Theorem 9. Suppose that the assumption 1)-3) hold. Suppose also that there ˜ K, K1 , K2 and exist integer j2 ≥ n and M1 , depending only on n, d1 , d, K11 , K12 . . . , such that |s · ω + ln+1 νn+1 (ω) + · · · + lj2 νj2 (ω)| ≥ K3 > 0

(8.9)

for all ω ∈ Ω, all integer n-vectors s and all j2 -vectors l such that |s| ≤ M1 and 1 ≤ |l| ≤ 2. Then for arbitrary γ > 0 and for sufficiently small ε ≤ ε¯(γ) (¯ ε > 0), a Borel subset Ωε ⊂ Ω and a Lipschitz embedding Σε : Tn × Ωε −→ Qδ can be found with the following properties: a) mes (Ω \ Ωε ) ≤ γ,

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S. B. Kuksin

b) the map Σε is Cερ -close to the map Σ0 : (q, ω) → (0, q, 0) ∈ Qδ and Lipschitz constant of the difference-map Σε − Σ0 is bounded by Cερ as well; c) each torus Σε (Tn × {ω}), ω ∈ Ωε , is invariant for the flow of equation (8.7) and is filled with its quasiperiodic solutions h(t) of the form h(t; q, ω) = Σε (q + ω t, ω), where ω = ω (ω) and |ω − ω| + Lip (ω − ω) ≤ Cερ . In statement b) of the theorem we view the difference Σε − Σ0 as a map, valued in the Hilbert space R2n × Yd . Amplification. Assertions b), c) hold with ρ replaced by one. Theorem 10. If in Theorem 9 all the frequencies νj are real, then a solution h(t) is linearly stable provided that the equation (8.7) linearised about this solution is uniformly well defined in the space R2n × Yd . For proofs these results with d1 ≥ 1, d˜ ≤ 0 see [K,P] and with d1 > 1 see [KK, K2]. 8.4

Completion of the Main Theorem’s proof (Step 4)

Now we apply Theorem 9 to equation (8.7) with Ω equal to a Borel subset Ωl ,l = {ω(r) | r ∈ R ,l }, l = 1, . . . , M , which we construct below, of the domain Ω and with γ = γ0 /(M C0 ), where the number C0 will be be chosen later. The assumptions 1)–3) hold with the constants from n through d11 , d21 , . . . the same as in Theorem 7, while the constants K and K2 depend on γ. We take j2 = j2 (γ) and M1 = M1 (γ) as in Theorem 9 and consider all resonances as in the l.h.s. of (8.9). Since the system of Floquet exponents {νj (r)} is nondegenerate, then each resonance does not vanish identically. As these functions are analytic, we can find K3 = K3 (γ) and for every l can ,l such that mes (Ω $ l \ Ωl ) ≤ γ/M and (8.9) holds for all find a subset Ωl ⊂ Ω ω ∈ Ωl . For every l we apply Theorem 9 with γ as above to find the subset Ωlε ⊂ Ωl , mes (Ωl \ Ωlε ) < γ0 , and the map Σlε : Tn × Ωlε −→ Qδ . $ε ⊂ R $ and the map Σε : R $ε × Now we are in position to define the set R Tn −→ Qδ , claimed in Theorem 7. We set: $ε = R

M

{r ∈ Rl | ω(r) ∈ Ωlε },

Σε (r, z) = G ◦ Shiftp ◦ Σlε (q(z), ω(r))

l=1

for r in Rl , where (r, z) → (p, q) is the action-angle transformation from Step 1. $ε and the map Σε satisfy all the claims of Theorem 7. Indeed, The set R $ε ) = mes (R $ \ ∪R) ˜ + mes (R \ R

M

,l | ω(r) ∈ Ω $ l \ Ωl } mes {r ∈ R

l=1

+

M l=1

,l | ω(r) ∈ mes {r ∈ R / Ωlε }.


117

$ε ) is bounded Denoting sup| det ∂ω/∂r|−1 by C(γ0 ) we see that mes (R \ R by γ0 + γ0 + C(γ0 )M γ0 /(M C0 ). This is smaller than 3γ0 , if C0 = C0 (γ0 ) is big enough. It means that we can choose γ = γ(ε) in such a way that $ ≤ 3γ0 goes to zero with ε and the assertion a) of Theorem 7 mes (R \ R) holds. The tori Σε ({r} × Tn ) are invariant for equation (8.3) and are filled with its quasiperiodic solutions of the form hε (t), where ωε = ω (p(r)). The estimates for Σε − Φ0 and ωε (r) − ω(r) readily follows from the corresponding estimates in Theorem 9. It remains to estimate Lipschitz constants of the differences as above. Let $ε × Tn . If r1 and r2 belong to us take any two points (r1 , z1 ) and (r2 , z2 ) in R $l , then the estimates for increments17 of Σε − Φ0 and ωε − ω the same set R follow from the corresponding estimates for the increments of Σ(0, q, 0) and ω −ω since the maps (8.16) are Lipschitz. If r1 and r2 belong to different sets $l , then |r1 − r2 | ≥ C(γ) > 0 and the increments of the differences divided R by the increments of the arguments is bounded by C1 ερ /C(γ). Since we can choose the rate of decaying γ(ε) → 0 to be as slow as we wish, then we can achieve C1 ερ /C(γ) ≤ C2 ερ˜, if we chose for ρ in Theorem 9 any number from the interval (˜ ρ, 1/3). The last arguments also show that the estimates |ω −ω| ≤ Cε and Lip (ω − ω) ≤ Cε imply that |ωε − ω| ≤ Cε and Lip (ωε − ω) ≤ Cρ ερ for any ρ < 1. It means that the Amplification to Theorem 9 implies the Amplification to Theorem 7. Finally, since linearisation of the symplectomorphism which sends solutions h(t) of (8.12) to solutions hε (t) transforms solutions of the corresponding linearised equations, then Theorem 8 follows from Theorem 10.

9

Examples

9.1

Perturbed KdV equation

Let us consider a perturbed KdV equation u˙ =

3 ∂ 1 uxxx + uux + ε fu (u, x) =: Vε (u)(x), 4 2 ∂x

(9.1)

under zero mean-value periodic boundary conditions. In (9.1) f (u, x) is a C d -smooth function (d ≥ 1), δ-analytic in u. Then the nonlinear part of the vector field Vε defines an analytic map of order one: H0d −→ H0d−1 , u −→ The equation’s hamiltonian is Hε = 17

3 ∂ uux + ε fu (u, x). 2 ∂x

2π 0

1 2 8u

! − 14 u3 − εf (u(x), x) dx.

i.e., the estimate |(ωε − ω)(r1 ) − (ωε − ω)(r2 ) ≤ Cερ |r1 − r2 |, etc.

118

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For ε = 0 this is the KdV equation and defined in (4.5) bounded part T 2n of any finite-gap manifold TV2n satisfies the assumptions i)-v) (see in Section 3). The linearised KdV equation has a system of Floquet solutions which is complete nonresonant (Section 6.2). The assumption 1) of Theorem 7 now holds with d1 = 3, d11 = · · · = 0 and 2) holds since dH = d˜ = 1. We get: Theorem 11. For any ρ < 1 and for sufficiently small ε > 0, there exists a Borel subset Rεn of the cube Rn = {0 < rj < K} and a Lipschitz map Σε : Rεn × Tn −→ H0d (S 1 ), analytic in the second variable, such that: a) mesn (Rn \ Rεn ) −→ 0 as ε → 0, b) the map Σε is ερ -close to the map Φ0 , Φ0 (r, z)(x) = G(Vx + z, r) (see (3.16 )), also in the Lipschitz norm, c) each torus Tεn (r) = Σε ({r} × Tn ), r ∈ Rεn , is invariant for equation (9.1) and is filled with its linearly stable time-quasiperiodic solutions of the form t → Σε (r, z + tωε (r)), where the n-vector ωε is Cε-close to W(r). To get the result we used Theorem 7, its Amplification and Theorem 8. The last theorem applies since the Floquet solutions of the linearised KdV equation have real exponents and since the linearised KdV equation is well posed. The theorem implies that the union of all linearly stable time-quasiperiodic solutions becomes infinite-dimensional and dense in H0d asymptotically as ε → 0: Corollary. The space H0d contains a subset Qε filled with linearly stable time-quasiperiodic solutions of (9.1) such that its Hausdorff dimension tends to infinity when ε → 0 and for any fixed function v ∈ H0d we have: distH0d (v, Qε ) −→ 0

as

ε → 0.

(9.2)

Proof. We define Qε as a union of all non-empty sets Σε (Rεn × Tn ) = T$ε2n , corresponding to all n-gap manifolds T 2n , n = 1, 2, . . . . By Proposition 4, the set T$ε2n has positive 2n-dimensional Hausdorff measure when ε is small. Thus, dimH Qε −→ ∞. To prove (9.2) we note that for any µ > 0 one can find n ≥ 1 and an n-gap potential u(x) such that u − vk ≤ µ (this is a famous result of V.A.Marchenko, see [Ma], Theorem 3.4.3 and [GT], p.27). Then u equals to Φ0 (r, z) with some r ∈ R and z ∈ Tn . If ε is sufficiently small, then by the assertion a) of the theorem, there exists r1 ∈ Rε such that |r − r1 | ≤ µ. Using b), we get that u − Σε (r1 , z)k ≤ Cµ + ερ and (9.2) follows since µ > 0 can be chosen arbitrary small. Another immediate consequence of the theorem is the observation that the Its–Matveev formula (4.4) with corrected frequency vector W “almost solves” the equation (9.1) for all t:


119

Corollary. For any r ∈ Rn+ and any z ∈ Tn there exists an n-vector Wε (r) and a solution uε (t, x) of (9.1) in H0d such that sup uε (t, ·) − 2 t

∂2 ln θ(i(V · +Wε t + z); r)d −→ 0 ∂x2

as

ε → 0.

Proof. For every ε > 0 let us choose any rε ∈ Rε such that rε → r as ε → 0 and take uε (t) = Σε (rε , z + tωε (r)). Then uε (t) − Φ0 (r, z+ωε t)k ≤ uε (t) − Φ0 (rε , z + ωε t)k + Φ0 (rε , z + ωε t) − Φ0 (r, z + ωε t)k = o(1)

as ε → 0.

2

∂ This implies the result since Φ0 (r, z + ωε t)(x) = 2 ∂x 2 ln θ(i(Vx + Wε t + z); r), where Wε = ωε .

An easy analysis of the first step in the proof of Theorem 9 (see [KK]) shows that the new frequency vector W ε has the form W ε (r) = W (r) + εW 1 (r) + W 1!1are obtained by averaging O(ε2 ), where components W1j of the n-vector 0 ! 2π − 0 f (u, x) dx . Here along the torus T n (r) 18 of the function G∗ ∂p∂ j G : (p, q, y) → u(·) is the normal form transformation from Theorem 11. Therefore the assertion of the second Corollary can be viewed as an averag∂2 ing theorem: for most r and for all z the functions 2 ∂x 2 ln θ(i(Vx+Wε t+z); r) with W ε = W (r) + εW 1 (r) + O(ε2 ) approximate solutions of the perturbed KdV equation (9.1) for all t and x, where the n-vector W 1 is obtained by the averaging described above. Here “for most r” means “for all r outside a set whose measure goes to zero with ε”. 9.2

Higher KdV equations

Let us consider a perturbation of the l-th equation from the KdV-hierarchy: u˙ =

∂ (∇u Hl + ε∇u H1 ), ∂x

(9.3)

! 2π (l)2 u +higher-order terms with ≤ l −1 derivatives dx where Hl (u) = Kl 0 2π and H1 = 0 f (x, u, . . . , u(l−1) ) dx. The function f is assumed to be C d smooth in x, . . . u(l−1) and δ-analytic in u, . . . , u(l−1) . Since ∇u H1 =

l−1 j=0

(−1)j

∂j f (j) (x, . . . , u(l−1) ), ∂xj u

then arguing as in Example 1.1, we see that order 2l − 1. 18

∂ ∂x ∇u H1

with respect to the measure (2π)−n dq = (2π)−n dz.

is an analytic map of

120

S. B. Kuksin

Let us take a bounded part T 2n of any n-gap manifold. It is invariant for the l-th KdV-equation (equal to (9.3)ε=0 ). As an invariant subset, it satisfies the assumptions i)-iv). The linearised equation has a complete system of Floquet solutions (see in Section 6). Due to (4.6) this system is nonresonant. Now Theorem 7 applies to equation (9.3) since the assumption 1) holds with d1 = 2l + 1, d11 = · · · = 0 (see (6.20)) and 2) holds with dH = d˜ = 2l + 1. We see that most of n-gap solutions of the l-th KdV equation persist in the perturbed equation (9.3) with sufficiently small ε in the same sense as for the KdV equation. 9.3

Perturbed SG equation

Since the system of finite-gap solutions (4.7), (4.8) of the SG equation under the even periodic boundary conditions satisfy the assumptions i)-iv) (Section 4) and the system of Floquet solutions (6.23) is complete nonresonant, then Theorem 7 applies to the perturbed equation,

= εf (u, x), utt − uxx + sin u (9.4) u(t, x) ≡ u(t, x + 2π) ≡ u0 (−x), where f is C k -smooth function (k ≥ 1), analytic in u. Accordingly, for any n ≥ 1, most of finite-gap solutions (4.7), (4.8) persist as time-quasiperiodic solutions of (9.4). The persisted solutions are Lyapunov-stable if the open gaps for solutions (4.7) are sufficiently small. Otherwise, they in general are non-stable. Since it is unknown if the finite-gap solutions of (SG) jointly are dence in a function space, then we do not know if the persisted solutions of (9.4) are asymptotically dense as ε → 0 (cf. (9.2)). 9.4

KAM-persistence of lower-dimensional invariant tori of nonlinear finite-dimensional systems

Let R2N" be an euclidean space, given the usual symplectic structure, let T 2N = r∈R Trn be an analytic submanifold of R2N , diffeomorphic to R×Tn , R Rn , and H1 , . . . , Hn be commuting hamiltonians, as in Proposition 3 (so they are defined and analytic in the vicinity of T 2n and each torus Trn is invariant for every hamiltonian vector field VHj ). Let us take any hamiltonian – say, H1 . Then the vector field VH1 |T 2n has the form ωl (r)∂/∂zl and by Proposition 3 linearised equations have Floquet solutions with some frequencies νj (r), which are real analytic functions. Applying Theorem 7 we get that: Theorem 12. Let us assume that the following analytic functions do not vanish identically: l · ν(r) + s · ω(r),

l ∈ ZN −n , 1 ≤ |l| ≤ 2; s ∈ Zn .

(9.5)


121

Let h be an analytic function, defined in the vicinity of T 2n . Then most of the tori Trn persist as invariant n-tori of the perturbed Hamiltonian vector field VH1 +εh , 0 < ε ( 1, in the sense, specified in Theorem 7. The persisted tori are filled with quasiperiodic solutions with zero Lyapunov exponents. This reduction of the Main Theorem is much easier than the Main Theorem itself. Its claim remains essentially true under weaker assumptions: it is sufficient to check that only functions (9.5) with |l| = 1 do not vanish identically, see [Bour] (we note that under this weaker assumption the claim about Lyapunov exponents is not true).

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Analytic linearization of circle diffeomorphisms Jean-Christophe Yoccoz

1

Introduction

Let T = R/Z and r ∈ {0, +∞, ω} ∪ [1, +∞). We denote Diffr+ (T) the group of homeomorphisms of T of class C r and C r -isotopic to the identity (if r = 0 it is the group of homeomorphisms of T; if r ≥ 1, r ∈ R∗ \ N, it is the group of C r diffeomorphisms whose r-th derivative verifies a Hölder condition of exponent r − )r%; if r = ω it is the group of R-analytic diffeomorphisms). We denote Dr (T) the group of C r -diffeomorphisms f of the real line such that f − id is Z-periodic. One can embed R into Dω (T) as the subgroup of translations α ∈ R → Rα ∈ Dω (T)

where

Rα : x → x + α,

and one can embed the quotient T = R/Z into Diffω + (T) as the subgroup of rotations α ∈ T → Rα ∈ Diffω + (T)

where

Rα : x → x + α(mod1).

One has Diffr+ (T) " Dr (T)/{Rp , p ∈ Z}. We consider the usual C r topology on Dr (T) and Diffr+ (T). Rotations on T have very simple dynamics. Poincaré asked under which condition a given homeomorphism f of T is equivalent (in some sense, e.g.: measurably, topologically, smoothly, analytically, etc.) to some rotation. He also gave a first answer to this question, today known as Poincaré’s classification theorem: if the rotation number of ρ (f ) is irrational then f is combinatorially conjugated to the rotation Rρ(f ) (see Corollary 3.3). In Section 3.3 we explain the Denjoy theory [De]. Fixing the rotation number to be irrational Denjoy proved that if we add regularity to the homeomorphism f (actually f is C 1 and Df has bounded variation) then f is topologically conjugated to the rotation Rρ(f ) . This theorem doesn’t give us information about the regularity of the conjugacy h. Denjoy proved also that the hypothesis f ∈ C 1+BV cannot be relaxed too much, constructing examples of C 1 diffeomorphisms which are not topologically conjugated to a rotation (see Section 3.4). In [He1] there is a generalisation of this result: for each irrational α there exists a dense subset, in


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the space of C 1 diffeomorphisms whose rotation number is α, such that every element of this subset is not topologically conjugated to a rotation. The step to higher order differentiability for the conjugation h requires new techniques and additional hypotheses on the rotation number. The fundamental regularity criterion is the following: Let k ≥ r ≥ 1, f ∈ Diff k+ (T). f is C r -conjugated to a rotation (then to the rotation Rρ(f ) ) if and only if the family (f n )n≥0 is bounded in the C r topology. In [Ar1] it was proved that if the rotation number verifies a diophantine condition and if the analytic diffeomorphism f is close enough to a rotation, then the conjugation is analytic. At the same time examples of analytic diffeomorphisms, with irrational rotation number, for which the conjugation is not even absolutely continuous were given. In [He1] Arnol’d’s result was improved obtaining a global result: there exists a set A ⊂ (0, 1), with full Lebesgue measure, such that if the rotation number of the C ∞ diffeomorphism belongs to A, then it is C ∞ conjugated to a rotation. A similar result holds for finitely differentiable diffeomorphisms, but in this case the conjugacy is less regular: this phenomenon of loss of differentiability is typical of small divisors problems. In the case of finite differentiability, the following result is due to Y. Katznelson and D. Ornstein [KO1,KO2], see also [Yo1,KS,SK]. Theorem 1.1 Let f be a C k circle diffeomorphism, k ∈ R, k > 2, with rotation number α. Assume that α verifies the following diophantine condition: there exists c > 0 and τ ≥ 0 such that for all n ∈ N, n > 0 |e2πinα − 1| > c|n|−τ −1 . Then the homeomorphism h which conjugates f to Rα is of class C k−1−τ − for all > 0. If f is C 2 and α is a number of constant type (i.e. τ = 0) then h is absolutely continuous. The previous result shows that the loss of differentiability is at most 1 + τ + . In Section 4 we will present our results in the analytic case. We distinguish between local and global situation. In the local case one must assume that the rotation number α belongs to the set B of Brjuno numbers (see Section 2.4). We have two statements: in the first one we assume that the diffeomorphism is analytic and univalent on some “big” complex strip B∆ = {z ∈ C : |m z| < ∆}, in the second one that the diffeomorphism is analytic in some “thin” complex strip Bδ but it is very close to a rotation: Theorem 1.2 Let f ∈ Diff ω + (T), ρ (f ) = α ∈ B and assume that ∆ > 1 B (α) +c, where c is a universal constant. If f is analytic and univalent 2π in B∆ then there exists h ∈ Diff ω + (T), analytic in B∆ , with ∆ = ∆ − 1 B (α) −c, which conjugates f to R in B . α ∆ 2π

Analytic linearization of circle diffeomorphisms

127

Theorem 1.3 Let f ∈ Diff ω + (T), ρ (f ) = α ∈ B. Given δ > 0 there exists 0 (α, δ) such that if f is holomorphic in Bδ and there |f (z) − z − α| ≤ 0 −1 then there exists h ∈ Diff ω on B δ . + (T) such that f = h ◦ Rα ◦ h 2

Using the theorem (proven in [Yo2]) showing that the Brjuno condition is also necessary for the problem of linearization of germs of complex analytic diffeomorphisms of (C, 0), and following the construction described in [PM], which associates to each non-linearizable germ of diffeomorphism of (C, 0) a non-linearizable analytic diffeomorphism of T close to a rotation, one sees that the Brjuno condition is necessary also in our context. In the global case we introduce a condition (condition H, see Section 2.5) which is sufficient and in some sense also necessary, more exactely: Theorem 1.4 Let f ∈ Diff ω + (T) and let α be its rotation number. If α ∈ H, there exists h ∈ Diff ω (T) such that f = h◦Rα ◦h−1 . Moreover if α ∈ H there + ω exists f ∈ Diff + (T) with rotation number α, which is not C ω -linearizable. Theorems 1.2 and 1.3 will be proved in Sections 4.2 and 4.3 respectively. The first part of Theorem 1.4 will be proved in Section 4.5, using an extension of Denjoy’s theory described in Section 4.4. The last Section 4.6 is devoted to the construction of counterexamples showing that the assumption α ∈ H in the global theorem is also necessary.

2 2.1

Arithmetics Introduction

The aim of this section is to introduce some elementary but fundamental facts from the theory of approximation of irrational numbers by rationals. The most important tool we will use here is the continued fraction expansion of a real number (see Section 2.2 and [HW]). As it will be clear in what follows, the action of the modular group GL (2, Z) on R = R ∪ {∞} plays a fundamental role in the renormalization procedure we will use to study the linearization problem. To better understand this action one can introduce a fundamental domain [0, 1) for one of the two generators (the translation T ) and concentrate the attention to the inversion x → 1/x restricted to [0, 1). Then one translates back 1/x to the fundamental domain and iterates the process. This gives an expanding map on [0, 1), the Gauss’ map A (2.1). As A is more and more expanding as x → 0+ one obtains a “microscope” and the symbolic dynamics of this map gives rise to the continued fraction algorithm. The convergents of the continued fraction expansion of an irrational number give its best rational approximations in a very precise sense (see Lemma 2.1). The classical diophantine conditions are reviewed in Section 2.3. They can be recasted in terms of the speed of growth of various quantities related to the continued fraction expansion. This opens the way to introducing two sets of

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“badly approximable” irrational numbers which include diophantine numbers and which we will prove being the optimal sets for the linearization problem (respectively local for the set B, introduced in Section 2.4, and global for the set H, introduced in Section 2.5). 2.2

Continued Fractions

We consider the map: A : (0, 1) → (0, 1) defined by: A (x) =

1 213 − x x

(2.1)

To each α ∈ R \ Q, iterating A, we associate its continued fraction expansion as follows. Let (αn )n≥0 be defined by: α0 = α − )α%,

αn = An (α0 )

for n > 0

(2.2)

∀n ≥ 1

(2.3)

and (an )n≥0 by a0 = )α%

−1 αn−1 = an + αn

then 1

α = a0 + a1 +

1 ..

.+

1 an + αn

or shortly α = [a0 , a1 , . . . , an + αn ]. The nth -convergent is defined by pn = [a0 , a1 , . . . , an ] = a0 + qn

1 a1 +

(2.4)

1 ..

.+

1 an

It is easy to identify the numerator and denominator of the nth -convergent with the sequences (pn )n≥−2 and (qn )n≥−2 , defined by: p−2 = 0, p−1 = 1, and q−2 = 1, q−1 = 0, and Moreover α =

pn +pn−1 αn qn +qn−1 αn

pn = an pn−1 + pn−2 ,

for all n ≥ 0

qn = an qn−1 + qn−2 ,

for all n ≥ 0

qn α−pn and αn = − qn−1 α−pn−1 .


129

Let β−1 = 1

βn =

n

n

αj ≡ (−1) (qn α − pn )

∀n ≥ 0

(2.5)

j=0

then qn+1 βn + qn βn+1 = 1 an+1 βn + βn+1 = βn−1 and as an ≥ 1 for all n −1

(qn + qn+1 )

−1

< βn < (qn+1 )

.

(2.6)

We then have the following standard results: Lemma 2.1 (Best approximation) Let q ≥ 0, p ∈ Z, n ≥ 0; if qα − p is strictly between qn α − pn and qn+1 α − pn+1 then either q ≥ qn + qn+1 or p = q = 0. We refer to [HW] for the elementary proof. Corollary 2.2 Consider the set {qα − p; 0 ≤ q < qn+1 , p ∈ Z}, ordered as . . . < x−1 < 0 = x0 < x1 < . . . . Let xk = qα − p be an element of this set. Then

xk+1 = xk + βn if q < qn+1 − qn • if n is even xk+1 = xk + βn + βn+1 if q ≥ qn+1 − qn

xk−1 = xk + βn if q < qn+1 − qn • if n is odd xk−1 = xk + βn + βn+1 if q ≥ qn+1 − qn Proof. We consider only the case n is even, the odd case being similar. We write xk+1 = q α − p , 0 ≤ q < qn+1 . If q < qn+1 − qn , xk + βn = (q + qn )α − (p + pn ) belongs to the set. If we had xk+1 < xk + βn , both q ≥ q (xk+1 being between xk and xk + βn ) and q ≤ q (xk + βn being between xk+1 and xk+1 + βn ) would be impossible by Lemma 2.1, a contradiction. If q ≥ qn+1 − qn , xk + βn + βn+1 belongs to the set. If we had xk+1 < xk + βn + βn+1 , as q ≥ 0 > q − qn+1 , xk+1 would be between xk = (xk + βn+1 ) − βn+1 and (xk + βn+1 ) + βn and we would have by Lemma 2.1 q ≥ q + qn ≥ qn+1 , again a contradiction. Let us consider the standard action of P GL (2, Z) = GL (2, Z) /{±I} on R = R ∪ {∞} given by: a b αa + b for g = ∈ GL (2, Z) , g · α = . (2.7) αc + d c d Then we have

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Proposition 2.3 Two irrational numbers α and α belong to the same GL (2, Z)-orbit if and only if there exists n ≥ 0 and n ≥ 0 such that αn = αn . 2.3

Diophantine Conditions

Given τ ≥ 0, α ∈ R \ Q satisfies a diophantine condition of order τ (for short α ∈ CD (τ )) if there exists a constant c > 0 such that: % c p p %% % (2.8) %α − % ≥ 2+τ ∀ ∈ Q, q ≥ 1. q q q Proposition 2.4 Let τ ≥ 0 and α ∈ R \ Q then the following statements are equivalent 1. 2. 3. 4. 5.

α ∈ CD (τ); qn+1 = O qnτ +1 ; an+1 = O(qnτ ); −1 αn+1 = O(βn−τ ); −1 βn+1 = O βn−τ −1 ;

where (αn )n≥0 , (pn /qn )n≥0 , (βn )n≥−1 and (an )n≥1 are defined in Section 2.2. The proof is elementary and is left as an exercise. We say that α satisfies a diophantine condition if α ∈ CD = ∪τ ≥0 CD (τ ). For all τ , CD (τ ) is invariant for the action of GL (2, Z). For all τ > 0, CD (τ ) has full measure; on the other hand (R \ Q) \ CD is a Gδ dense subset of R and CD (0) has zero measure. 2.4

Brjuno function and condition B

Let α ∈ R \ Q, we define (following [Yo2] and [MMY1]) the Brjuno function B : R \ Q → R+ ∪ {+∞} B (α) = βn−1 log αn−1 (2.9) n≥0

where (αn )n≥0 and (βn )n≥−1 are defined in Section 2.2. We say that α ∈ R \ Q satisfies the condition B (or is a Brjuno number, for short α ∈ B) if B (α) < +∞. Remark 2.5 As in [Yo2] and [MMY1] we extend the above definition to rational values setting B (α) = +∞ or e− B(α) = 0 for α ∈ Q. The following proposition collects some properties of the Brjuno function (see [MMY1], Proposition 2.3, p. 277, for a proof). Proposition 2.6 The function B satisfies


131

1. B (α) = B (α + 1) for all α ∈ R; 2. for all α ∈ (0, 1) then B (α) = log α−1 + α B α−1 ; 3. there exists a constant C1 > 0 independent of α such that for all α ∈ R \ Q % log qj+1 %% % (2.10) % ≤ C1 % B (α) − qj j≥0

where (qj )j≥0 are the denominators of the convergent of α defined in (2.4). Remark 2.7 By 2. one has that B is P GL (2, Z)-invariant. Indeed the Brjuno function B is a P GL (2, Z)-cocycle (see [MMY2], Appendix 5). Remark 2.8 Let α ∈ CD, by Proposition 2.4 there exists τ ≥ 0 such that qn+1 = O qnτ +1 , thus logqqnn+1 ≤ (1 + τ ) logqnqn + qcn and then B (α) < +∞. Therefore CD ⊂ B and the set B has full measure. Remark 2.9 For positive τ we could define a condition CD (τ ) using the − τ1 function B (τ ) (α) = (note that the function B (τ ) (α) = j≥0 βj−1 αj −1 τ would have the same features) as follows: j≥0 βj−1 αj α ∈ CD (τ ) if and only if B (τ ) (α) < +∞. Then the condition CD (τ ) is almost equivalent to condition CD (τ ), in fact CD (τ ) ⇒ CD (τ ) , CD (τ ) ⇒ CD (τ ) 2.5

∀τ > τ.

Condition H

For α ∈ (0, 1), x ∈ R, define

α−1 (x − log α−1 + 1) if x ≥ log α−1 , rα (x) = ex if x ≤ log α−1 .

(2.11)

Observe that rα is of class C 1 on R, satisfying rα (log α−1 ) = Drα (log α−1 ) = α−1 ; ex ≥ rα (x) ≥ x + 1 for all x ∈ R; Drα (x) ≥ 1 for all x ≥ 0. For α ∈ R \ Q, we now set, for k > 0 ∆k (α) = rαk−1 ◦ · · · ◦ rα0 (0). An easy calculation gives

(2.12) (2.13) (2.14)

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Lemma 2.10 Let α ∈ B and k ≥ 0. Assume that B(αk ) ≤ ∆k (α). Then one has, for l ≥ k βl−1 ∆l (α) = βk−1 ∆k (α) +

l−1

βj−1 −

j=k

= βk−1 (∆k (α) − B(αk )) +

l−1

βj−1 log αj−1

j=k l−1

βj−1 + βl−1 B(αl )

j=k

≥ βl−1 B(αl ) ≥ βl−1 log αl−1 . Let us define, for n ≥ k ≥ 0: Hk,n = {α ∈ B, B(αn ) ≤ ∆k (αn−k )}, and then H = ∩m≥0 (∪k≥0 Hk,k+m ) = {α ∈ B, ∀m ≥ 0, ∃k ≥ 0, B(αm+k ) ≤ ∆k (αm )} Proposition 2.11 1. For 0 < k ≤ n, one has Hk−1,n ⊂ Hk,n ⊂ Hk+1,n+1 ; 2. Let α ∈ R \ Q. Then α ∈ H (resp. α ∈ B) if and only if the orbit of α under GL (2, Z) is contained in (resp. intersects) ∪k≥0 Hk,k . 3. The subset H of R is a Fσδ subset (i.e. a countable intersection of Fσ subsets). Proof. 1. From Lemma 2.10 we have Hk,n ⊂ Hk+1,n+1 . Also ∆1 (αn−k ) = 1 and from (2.14) it follows that ∆k (αn−k ) ≥ ∆k−1 (αn−k+1 ) + 1 and thus Hk−1,n ⊂ Hk,n . 2. Let α ∈ R \ Q. Then α ∈ H if and only if αn ∈ ∪k≥0 Hk,k for all n ≥ 0. As the αn ’s belong to the orbit of α under GL (2, Z), we see that if the orbit of α is contained in ∪k≥0 Hk,k then α ∈ H. Conversely, if α ∈ H and α belongs to the orbit of α, by Proposition 2.3 there exist integers n and n such that αn = αn . Then αn ∈ ∪k≥0 Hk,k and it follows from 1. that also α ∈ ∪k≥0 Hk,k . On the other hand, we have by definition ∪k≥0 Hk,k ⊂ B, and B is invariant under GL (2, Z). Conversely, if α ∈ B, let k ≥ B(α) and α ∈ R \ Q be such that αk = α0 . Then α belongs to the orbit of α and we have by (2.13) ∆k (α ) ≥ k ≥ B(αk ) hence α ∈ Hk,k . 3. As B is lower semicontinuous, for every n ≥ k ≥ 0, Hk,n is a closed subset and the conclusion follows. Remark 2.12 One has the obvious inclusions CD ⊂ H ⊂ B. Moreover, by 2. H is GL (2, Z)-invariant. Example 2.13 Consider the compact subset K ⊂ [0, 1] whose elements are irrationals α such that a1 = 1,

ai+1 ≤ exp (2ai ) − 1.


133

Let α ∈ K, j ≥ 0; denote by pñ /˜ qn the convergents of αj . For n > 0, we have   −1 −1  αj+l  log αj+n ≤ qñ−1 log(aj+n+1 + 1) ≤ 2˜ qn−1 aj+n ≤ 2˜ qn−1 ; 0≤l log α−1 ; as x < rα (x) < ex , the number t = L(rα (x)) − L(x) belongs to (0, 1). Consider the map w → Et (w) − rα (w) for w ≥ log α−1 . It is convex, negative at log α−1 , and vanishes at x; therefore the derivative at x is positive, and thus we must have D(L ◦ rα − L)(x) < 0. Lemma 2.15 Let α ∈ B. 1. For any k ≥ 0, we have B(αk+1 ) ≤ exp(B(αk ) − 1). 2. If α ∈ / ∪k≥0 Hk,k , the positive sequence L(B(αk )) − L(∆k (αk )) is decreasing. Proof. 1. The assertion follows immediately from the relation B(αk+1 ) = eu (B(αk ) − u),

u = log αk−1

maximizing over u. 2. From the first assertion we have always B(αk+1 ) ≤ rαk (B(αk ) − 1) ≤ rαk (B(αk )). Therefore the second assertion of Lemma 2.15 follows from the second assertion of Lemma 2.14. We now consider an irrational number α ∈ B. Clearly, if α ∈ / H, we must have lim B(αk ) = +∞.

k→∞

Fix c0 > 0 large enough such that, for any irrational α, ˜ 1/2 ≤ c0 . β˜ k−1

k≥0

We define, starting with k0 = 0, an increasing sequence of indices ki setting 1/2 βki ki+1 = inf{k > ki , log αk−1 ≥ c−1 B(αki +1 )}. 0 βk−1


135

The set on the right-hand side is indeed nonempty: otherwise, as B(αki +1 ) =

βk−1 log αk−1 βki

k>ki

we would have

B(αki +1 ) < c−1 0

βk−1 1/2 B(αki +1 ) ≤ B(αki +1 ). βki

k>ki

Lemma 2.16 For α ∈ B and ki as above we have ) = 0. lim L(B(αki )) − L(log αk−1 i

i→+∞

If α ∈ /H lim L(B(αki +1 )) − L(B(αki )) = 1.

i→+∞

Proof. For the first limit, we note that βki βki+1 −1

B(αki +1 ) ≥ B(αki+1 ) ≥

log αk−1 i+1

≥

c−1 0

βki

1/2 B(αki +1 )

βki+1 −1

(2.15) But it is easily checked that lim

x≥1,y≥1,xy→∞

1/2 L(xy) − L(c−1 y) = 0 0 x

from which the assertion follows. The second part of the Lemma is then an immmediate consequence of the second part of Lemma 2.15: as α ∈ / H the sequence L(B(αk )) − L(∆k (α)) is positive decreasing; setting εi = L(B(αki )) − L(log αk−1 ), we have that, if i L(∆ki (α)) ≤ L(B(αki )) − εi , then L(∆ki +1 (α)) = L(∆ki (α)) + 1 ≤ L(B(αki +1 )). This forces the second assertion of the Lemma.

We now consider the two relations (2.15) and βki 1 1 1 = βki+1 −1 βki+1 −1 βki −1 αki

.

We get 4 lim

L(βk−1 ) i+1 −1

− max

L(βk−1 ), L(αk−1 ), L i −1 i

β ki βki+1 −1

5 =0

136

and

J.-C. Yoccoz

lim L(B(αki+1 )) − max L 4

βki βki+1 −1

5 , L(B(αki +1 ))

=0

which, with Lemma 2.16, transform to 4 5 β ki −1 −1 −1 0 = lim L(βki+1 −1 ) − max L(βki −1 ), L(log αki ) + 1, L βki+1 −1 4 5 βk i −1 −1 = lim L(log αki+1 ) − max L , L(log αki ) + 1 . βki+1 −1 We claim that Proposition 2.17 We have limi→+∞ L(log αk−1 ) − L(βk−1 ) = 0. i+1 i+1 −1 Proof. If not we would have ) > L(log αk−1 )+1 L(βk−1 i −1 i for infinitely many i’s. But setting β ki −1 −1 ) − max L(β ), L(log α ) + 1, L εi = L(βk−1 ki −1 ki i+1 −1 βki+1 −1 βk i −1 εi = L(log αk−1 , L(log α ) − max L ) + 1 ki i+1 βki+1 −1 ηi = L(βk−1 ) − L(log αk−1 ) i −1 i we have that if ηi+1 > εi − εi then −1 −1 max L(βki −1 ), L(log αki ) + 1, L and L(βk−1 ) i −1

ηi+1 = εi −

εi

+

≤ εi −

εi

+ ηi − 1.

− max L

β ki βki+1 −1 βk i

βki+1 −1

= L(βk−1 ) i −1

, L(log αk−1 ) i

+1

Corollary 2.18 Let α ∈ / H. Then for any t < 1 there are infinitely many rationals such that |α − p/q| ≤ (Et (q))−1 . 2.6

Z2 -actions by translations and continued fractions

Let (e−1 , e0 ) be the canonical basis of Z2 . We consider the following action π Z2 → Homeo+ (R): to (e−1 , e0 ) we associate two commuting homeomorphisms by

π(e−1 ) : x → x − 1 = T (x) (2.16) π(e0 ) : x → x + α = f (x)


137

We perform the sequence of changes of basis of Z2 given by

en−1

en

e−1

= Mn

,

e0

where

Mn =

pn−1 qn−1 pn

,

qn

thus

en

en+1

= An+1

en−1 en

,

where

An+1 =

0

1

1 an+1

.

The homeomorphisms corresponding to the new basis (en−1 , en ) will be (π(en−1 ), π(en )) where π(en ) :

x → x + (−1)n βn .

We can now rewrite Lemma 2.1 (from now on we will use the multiplicative notation for Zk -actions): Lemma 2.19 (Best approximation 2nd version) Let p, q ∈ Z, if qα − p is strictly between en · 0 and en−1 · 0 then either |q| ≥ qn + qn−1 or p = q = 0.

3 3.1

The C r theory for r ≥ 0 The C 0 theory

We consider an action of Zk on R, i.e. an homomorphism π : Zk → Homeo+ (R). We will assume that the action has no common fixed point. We define Z + = {e ∈ Zk : e · x > x ∀x ∈ R}; we have then −Z + = {e ∈ Zk : e · x < x ∀x ∈ R}.

Proposition 3.1 The subset Z + of Zk has the following properties: 1. 2. 3. 4. 5.

Z + = ∅; −Z + ∩ Z + = ∅; if e ∈ −Z + and e ∈ Z + then e + e ∈ Z + ; for any r ≥ 1, e ∈ Z + if and only if re ∈ Z + ; for any e ∈ Z + , e ∈ Zk there exists n such that e + ne ∈ Z + .

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J.-C. Yoccoz

Proof. 1. We prove by induction on k that if Z + = ∅ the action has a common fixed point. Write e1 , . . . , ek for the canonical basis of Zk . When k = 1, e1 must have a fixed point if neither e1 nor −e1 belong to Z + . From the induction hypothesis, we know that the closed set F of points fixed by e1 , . . . , ek−1 is non empty; also ek has a fixed point x0 . If x0 ∈ F , we are done; otherwise, as ek commutes with ei , i < k, the set F is ek -invariant, and the component of R \ F containing x0 is fixed by ek . But then an endpoint of this component in F is fixed by ek . 2. Obvious 3. By assumption, there exists x0 such that e · x0 ≥ x0 . As e ∈ Z + , and thus has no fixed point, there exists for any x ∈ R an integer n ∈ Z such that ne · x0 ≤ x < (n + 1)e · x0 . Then (e + e ) · x ≥ [(n + 1)e + e ] · x0 ≥ (n + 1)e · x0 > x. 4. Obvious 5. Let x0 ∈ R; as above there exists n ∈ Z such that (n − 1)e · (e · x0 ) ≥ x0 . Then (n − 1)e + e ∈ / −Z + and ne + e ∈ Z + by 3. Remark 3.2 A subset Z + ⊂ Zk satisfies the properties 1–5 of Proposition 3.1 if and only if there exists a non zero linear form ρ : Zk → R such that Z + = {e ∈ Zk , ρ(e) > 0}, the form ρ being then uniquely determined up to a positive scalar multiple. We leave the proof as an exercise. The form ρ may be defined as follows. Choose e0 ∈ Z + and define, for e ∈ Zk N + (e) = inf{n ∈ Z, ne0 − e ∈ Z + } N − (e) = sup{n ∈ Z, ne0 − e ∈ −Z + } Then N + (ke) N − (ke) = lim . k→∞ k→∞ k k

ρ (e) = lim

(3.1)

Fix such a linear form ρ and consider the action by translations πρ (e) · x = x + ρ(e),

for e ∈ Zk , x ∈ R.

Given x0 ∈ R, ρ vanishes on the π-stabilizer of x0 ; this allows to define a map h from the π-orbit of x0 onto ρ(Zk ) by h(e · x0 ) = ρ(e). This map is immediately seen to be order preserving. When the Z-rank of ρ is > 1, ρ(Zk ) is dense in R, and h has a unique order preserving extension to R, which is automatically continuous and surjective and still a semi-conjugacy from π to πρ . We thus have


139

Corollary 3.3 (Poincaré) When the Z-rank of ρ is > 1, there is an order preserving surjective continuous semi-conjugacy h from π to πρ : h(e · x) = h(x) + ρ(e), ∀x ∈ R, ∀e ∈ Zk uniquely determined up to an additive constant. 3.2

Equicontinuity and topological conjugacy

Let F be an element of Diffr+ ( T) and f ∈ Dr (T) a lift of F . To such an F we can associate the Z2 -action π defined as follows: to the canonical basis of Z2 we associate

f−1 : x → x − 1 = T (x) , (3.2) : x → f (x) . f0 Then ρ (π) = (−1, α) and the number α ∈ R is called the rotation number of f (or associated to the Z2 -action π; we will also write α = ρ(f ) and we define the rotation number of F being ρ(F ) = α(mod1)). Poincaré’s result (Corollary 3.3) can be recasted in the following form: Proposition 3.4 (Poincaré) Let f ∈ D0 (T), α = ρ(f ) ∈ R \ Q. There exists h : R → R continuous, monotone non-decreasing such that h−id is Z-periodic and h ◦ f = Rα ◦ h. Remark 3.5 By Proposition 3.4 µ = Dh is an F -invariant probability measure on T and h is a homeomorphism if and only if the support of µ is T1 . Let s ∈ {0, +∞, ω} ∪ [1, +∞), s ≤ r. The fundamental criterion of C s conjugacy of a diffeomorphism F to the rotation Rα , where α = ρ(F ), has been stated in the Introduction. Here, following [He1] (Section II.9) we will prove a criterion that guarantees that F is topologically conjugated to Rα . For the general case we refer to [He1] (Chapitre IV). Proposition 3.6 Let F ∈ Diffr+ (T), α = ρ(F ). F is C 0 conjugated to Rα if and only if the family (F ni )i∈N is equicontinuous for some ni → ∞. Proof. The condition is clearly necessary: here we assume that α ∈ R \ Q and we prove that it is also sufficient. We leave to the reader the case α ∈ Q. Let µ be an F -invariant probability measure. By Remark 3.5, F is topologically conjugated to Rα if and only if suppµ = T1 . If one has K = suppµ = T1 , since K is F -invariant, T1 \ K is open and F -invariant. Let J be a connected component of T1 \ K. Then the intervals (F n (J))n∈Z are pairwise disjoint connected components of T1 \ K. Thus |F n (J)| → 0 as n → ±∞, but this contradicts the equicontinuity of (F n )n∈N .

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3.3

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The Denjoy theory.

Let F be an element of Diff 2+ (T) and f a lift of F . Set k = 2 and consider π We assume that the rotation the action Z2 → Diff 2+ (R) defined as in (3.2). ! pn will be the convergents of α number α ∈ R \ Q ∩ (0, 1). As usual qn n≥0

and we can define a new basis of Z2 by (fn−1 , fn ) where fn = f qn T pn . The goal of this section is to prove Denjoy’s Theorem (Corollary 3.13): if the action is by C 2 diffeomorphisms then the semiconjugacy h of Corollary 3.3 is indeed a conjugacy (there is no interval on which h is constant). The fundamental step will be Corollary 3.12 which allows to obtain a bound for |Dfn |C 0 . The following lemma is fundamental for all the theory, in fact it gives control of the reciprocal position of the iterates of a fixed interval. Lemma 3.7 Let In (x) = [x, fn (x)] and Jn (x) = In (x) ∪ In fn−1 (x) = [fn−1 (x), fn (x)]. Then 1. for 0 ≤ j < qn+1 and k ∈ Z the intervals f j T k (In (x)) have disjoint interiors; 2. for 0 ≤ j < qn+1 and k ∈ Z the intervals f j T k (Jn (x)) cover R (at most twice). Proof. We assume for instance that n is even. Since the two statements are invariant under conjugacy, we can assume that f is the translation by α and x = 0. With the notations of Corollary 2.2, write f j T k (0) = xl ; then f j T k (In (0)) = (xl , xl + βn ) and f j T k (Jn (0)) = (xl − βn , xl + βn ). Therefore both statements in the Lemma follow from Corollary 2.2. Remark 3.8 We note that in what follows the hypothesis C 2 can be replaced by f ∈ C 1+BV , i.e. f is C 1 and Df has bounded variation. In fact we will always need control of variation of the function log Df (or of something similar) when Df is positive1 . The following proposition shows that using Lemma 3.7 we can bound the variation of log Df j on the intervals In (x) with the total variation of log Df on [0, 1]. Using Proposition 3.9 we then prove Corollaries 3.10 and 3.11, which, together with Corollary 3.12, imply Denjoy’s Theorem (Corollary 3.13). Proposition 3.9 Let f ∈ C 2 then for 0 ≤ j ≤ qn+1 Var logDf j ≤ Var logDf = V In (x) 1

(3.3)

[0,1]

If ψ has bounded variation and φ is Lipschitz with constant Lφ then Var (φ◦ψ) ≤ Lφ Var ψ


141

Proof. The chain rule applied to f j gives: log Df j =

j−1

log Df ◦ f i .

(3.4)

i=0

The estimate (3.3) then follows from the first assertion of Lemma 3.7. Proposition 3.9 has the following corollaries: Corollary 3.10 Let f, V, n and j be as in Proposition 3.9. Then |In f j (x) | −V ≤ eV . ≤ e |In (x)|Df j (x) Proof. We first note that In f j (x) = f j (In (x)) and secondly that

(3.5)

|f j (In (x))| ≤ |In (x)|Df j (ξ) for some ξ ∈ In (x). Thus by (3.3) and Lemma 3.7 one has % % |In f j (x) | %% %% % % j j ≤ log Df (ξ) − log Df (x) % % % ≤ V. % log |In (x)|Df j (x)

Corollary 3.11 Let f, V, n and j be as in Proposition 3.9. Then |log Dfn ◦ f j (x) − log Dfn (x)| ≤ V

(3.6)

Proof. Since fn and f j commute we can write log Dfn ◦ f j + log Df j = log Df j ◦ fn + log Dfn .

(3.7)

On the other hand, it follows from Proposition 3.9 that |log Df j ◦ fn − log Df j | ≤ V.

(3.8)

Corollary 3.12 (Denjoy’s inequality) Let f and V be as in Proposition 3.9. Then |log Dfn |C 0 ≤ 2V

(3.9)

Proof. Since Dfn is a periodic function with mean value 1, there exists x0 ∈ R such that Dfn (x0 ) = 1, i.e. log Dfn (x0 ) = 0. Given x ∈ R, take 0 ≤ j < qn+1 and k ∈ Z such that x1 = f j T k (x0 ) ∈ Jn (x). Then we have: |log Dfn (x1 )| = |log Dfn (x1 ) − log Dfn (x0 )| ≤ V,

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and since either2 x1 ∈ In (x) or x ∈ In (x1 ), we also have |log Dfn (x) − log Dfn (x1 )| ≤ V. We can then conclude: |log Dfn (x)| ≤ |log Dfn (x) − log Dfn (x1 )| + |log Dfn (x1 ) − log Dfn (x0 )| ≤ 2V.

Corollary 3.13 (Denjoy’s Theorem) All C 2 Z2 -actions (3.2) with irrational rotation number are topologically conjugated to the standard action (2.16) by translations of (−1, α). Proof. This follows from Proposition 3.6 as Denjoy’s inequality imply that the (fn )n≥0 are equicontinuous. 3.4

Denjoy’s counter-examples

Denjoy has constructed examples of C 1 diffeomorphisms with no periodic orbits but not topologically conjugated to a rotation [De]. Indeed one can actually prove that Theorem 3.14 For every irrational α, there exists a diffeomorphism f of class C 2−ε for all ε > 0, with rotation number equal to α which is not topologically conjugated to a rotation. We can find similar results in [DMVS,He1]. 3.5

The Schwarzian derivative

To get a more precise control of the non-linearity of fn we introduce the Schwarzian derivative S. Let F ∈ Diff 3+ (T) and f a lift of F , then the the Schwarzian derivative of f is defined by 3 1 D3 f 2 Sf = D log Df − (D log Df ) = − 2 Df 2 2

D2 f Df

2 (3.10)

The following proposition collects some properties of S. Proposition 3.15 Let f ∈ Diff 3+ (R) then 2

1. S (f ◦ g) = (Sf ) ◦ g (Dg) + Sg; 2 n−1 2. for n ≥ 1 then Sf n = i=0 (Sf ) ◦ f i Df i ; √ 3. |D log Df |C 0 ≤ 2 |Sf |C 0 . 2

In fact if x1 ∈ In (x) then x1 ∈ fn−1 (x) , x , from which it follows that x1 ≤ x ≤ fn (x1 ), i.e. x ∈ In (x1 ).


143

Proof. 1. is immediate and 2. can be checked by induction using 1. To prove 3. choose x0 such that |D log Df |C 0 = |D log Df (x0 )|, i.e. D2 log Df (x0 ) = 0. Then |D log Df (x0 )| = 2|Sf (x0 )| from which 3. follows. Corollary 3.16 Let f ∈ Diff 3+ (R), S = Sf C 0 and Mn = fn − idC 0 . For 0 ≤ j ≤ qn+1 one has |Sf j (x)| ≤

Mn e2V S |In (x)|2

Proof. Using 2. of Proposition 3.15 we get j−1 % 2 %% % |Sf j (x)| = % (Sf ) ◦ f i (x) Df i (x) % i=0

≤S

j−1

|Df i (x)|2

i=0

Using Corollary 3.10 we obtain: j−1 Se2V i |In f (x) |2 . 2 |In (x)| i=0 2 Since Mn = maxx |In (x)|, the trivial ai ≤ supi |ai | |ai |, and i inequality the disjointness of the intervals In f (x) lead to the desired conclusion.

|Sf j (x)| ≤

Corollary 3.17 Let f ∈ Diff 3+ (R), mn = minx |In (x)| and Mn = maxx |In (x)|. For 0 ≤ j ≤ qn+1 one has 1

|D log Df | j

with c1 =

C0

√ 2SeV .

c1 Mn2 ≤ mn

Proof. Using 3. of Proposition 3.15 and Corollary 3.16 we have √ 1 √ √ (Mn ) 2 eV S |D log Df j |C 0 ≤ 2 |Sf j |C 0 ≤ 2 . minx |In (x)| Corollary 3.18 Let f ∈ Diff 3+ (R), mn = minx |In (x)| and Mn = maxx |In (x)|. For 0 ≤ j ≤ 2qn+1 one has 1

|D log Df |C 0 j

with c2 =

√ 2SeV 1 + e2V .

c2 Mn2 ≤ mn

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Proof. We only need to consider the case qn+1 ≤ j ≤ 2qn+1 , then i = j−qn+1 , varies between 0 and qn+1 . Applying Corollary 3.17 to D log Df i ◦ fn+1 the thesis follows. Corollary 3.19 Let f ∈ Diff 3+ (R), Mn = maxx |In (x)|. For m = n, n + 1 one has 1

|D log Dfm (x)| ≤ where c3 =

c3 Mn2 |In (x)|

√ 2Se2V 1 + e2V 2 + e2V .

Proof. Let x0 ∈ R be such that mn = |In (x0 )|, and let x ∈ R. Choose 0 ≤ j < qn+1 , k ∈ Z, y ∈ Jn (x0 ) such that x = f j T k (y). One has D log D fm ◦ f j (y) = D log Dfm (x) Df j (y) + D log Df j (y) , and therefore, by Corollaries 3.17, 3.18: √ |D log Dfm (x)| ≤ 2SeV 2 + e2V |Df j (y)|−1 . On the other hand, from Corollary 3.10, one has |Df j (y)|−1 ≤

|In (y)| V e . |In (x)|

Finally, we have by Denjoy’s inequality |In (fn (x0 ))| ≤ e2V |In (x0 )|, |In (fn−1 (x0 ))| ≤ e2V |In (x0 )|, and therefore, as In (y) ⊂ In (x0 )∪In (fn (x0 )) or In (y) ⊂ In (x0 )∪In (fn−1 (x0 )): |In (y)| ≤ 1 + e2V |In (x0 )| which proves the required inequality. We can finally prove 1

Corollary 3.20 Let f as in Proposition 3.15. Then |log Dfn |C 0 ≤ c4 Mn2 . Proof. Let x0 ∈ T such that mn = |In (x0 )| (so log Dfn (x0 ) = 0); from Lemma 3.7 we know that Jn (x0 ) cover the circle, so for all x ∈ T there exists 0 ≤ j ≤ qn+1 and j ∈ Jn (x0 ) such that x = f j (y). Letting 0 ≤ j ≤ 2qn+1 we can take y ∈ In (x0 ). We use the fact that log Df j (fn (y)) − log Df j (y) = log Dfn (x) − log Dfn (y)


145

we write log Dfn (x) − log Dfn (x0 ) = log Df j (fn (y)) − log Df j (y) + log Dfn (y) − log Dfn (x0 )

(3.11) (3.12)

To estimate the right hand side of (3.11) we use |log Df j (fn (y)) − log Df j (y)| ≤ |

fn (y)

D log Df j (ξ) dξ|

y

≤ mn (y) ||D log Df j ||C 0 ≤

1 |In (y)| c2 Mn2 |In (x0 )|

but |fn (y) − y| ≤ |fn (y) − fn (x0 )| + |x0 − y| ≤ (eV + 1)mn (x0 ). For (3.12) we use something similar. And finally we obtain the result. 3.6

Partial renormalization

In this subsection we prove a first step of every renormalisation theory for circle diffeomorphisms. We prove that starting with some f ∈ Diff ω + (R), not necessarily near a translation, then for all > 0 we can do an appropriate number of renormalisation steps in order to end with an analytic diffeomorphism f which is close to a translation in the C 2 topology. Proposition 3.21 Let us consider the action π : Z2 → Diff ω + (R), given by: π : e−1 → π (e−1 ) = T π : e0 → π (e0 ) = f with ρ (π) = [−1, α] and α ∈ R \ Q. Let fn = f qn ◦ T pn and (fn , fn+1 ) be a basis of the Z2 -action. Let > 0, then for n large enough, there exists hn ∈ Diff ω + (R) such that: hn ◦ fn ◦ h−1 n =T hn ◦ fn+1 ◦ h−1 n = Fn+1 with Fn+1 ∈ Diff ω + (R), ρ (Fn+1 ) = αn+1 and |D log DFn+1 |C 0 < . Proof. Let x0 ∈ T be such that mn = |In (x0 )| and let A be the affine map such that A(x0 ) = 0, A(fn (x0 )) = −1. Set fˆm = A◦fm ◦A−1 for m = n, n+1. By Corollaries 3.17 and 3.20, we have log Dfˆm C 1 ≤ cMn1/2 . Moreover fˆn (0) = −1; therefore there exists k0 ∈ Diff 2+ ([−1 , 1 ]) with k0 − 1/2 idC 2 < cMn and k0 ◦ fˆn ◦ k0−1 = T on [0, 1]. On the other hand, the

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quotient R/fˆnZ is a compact analytic 1-dimensional manifold, and is therefore ˆ C ω -diffeomorphic to the circle: there exists k1 ∈ Diff ω + ([−1 , 1 ]) with k1 ◦ fn ◦ k1−1 = T on [0, 1]. 2 The composition k0 ◦ k1−1 belongs to Diff 2+ (T); taking k ∈ Diff ω + (T), C ω −1 close to k0 ◦ k1 , and setting k2 = k ◦ k1 , we have k2 ∈ Diff + ([−1 , 1 ]) and 1/2 k2 − idC 2 < cMn . ˆ −1 It is now sufficient to take hn = k2 ◦A: we have hn ◦fn ◦h−1 n = k2 ◦ fn ◦k2 = −1 −1 ˆ T and Fn+1 := hn ◦ fn+1 ◦ hn = k2 ◦ fn+1 ◦ k2 , with log DFn+1 C 1 ≤ cMn1/2 .

4 4.1

Analytic case A linearization criterion

Let π : Z2 → Diff ω + (R) be an action with π(e−1 ) = T , π(e0 ) = f as above. Let ∆ > 0 be such that f is holomorphic and univalent in B∆ . Proposition 4.1 The action π is C ω -linearizable if and only if the set 6 L= f −n (B∆ ) n≥0

contains R in its interior. Proof. The condition is obviously necessary. Assume that it is satisfied. Then the component U of intL which contains R is invariant under T and satisfies f (U ) ⊂ U . It is simply connected by the maximum principle (applied to m f n ); therefore there exists δ > 0 and a real biholomorphism h : Bδ → U commuting with T . Then h−1 ◦ f ◦ h is real, commutes with T and sends univalently Bδ into itself. Therefore it is a translation. 4.2

Local Theorem 1.2: big strips

2 4.2.1 Scheme of the proof Let π : Z2 → Diff ω + (R) be an action of Z with generators π(e−1 ) = T , π(e0 ) = f . We assume that ρ(f ) = α is a Brjuno number and that f is holomorphic and univalent in a strip B∆ with 1 ∆ > 2π B(α) + c, where c is a conveniently large universal constant. We set F0 = f , ∆0 = ∆, α0 = α. We will denote by c1 , c2 , c3 universal constants. We will construct, through a geometric construction explained in the next section, holomorphic maps Hn , Fn for n > 0 with the following properties:

• (in ) Fn ∈ Diff ω + (R) commutes with T , has rotation number αn , and is holomorphic and univalent in a strip B∆n with


147

1 • (iin+1 ) ∆n+1 = αn−1 ∆n − 2π log αn−1 − c1 . • (iiin+1 ) The map Hn+1 is holomorphic and univalent in the rectangle Rn = 1 {| e z| < 2, | m z| < ∆n − 2π log αn−1 − c2 }. Moreover Hn+1 restricted to (−2, 2) is real and orientation reversing. • (ivn+1 ) For z, z ∈ Rn we have |αn (m Hn+1 (z) − m Hn+1 (z )) + (m z − m z )| ≤ c3 . • (vn ) For z ∈ Rn we have 11 9 αn < e(Fn (z) − z) < αn 10 10 1 αn | m(Fn (z) − z)| < 10 • (vin+1 ) For z ∈ Rn ∩ T −1 Rn Hn+1 (T z) = Fn+1 (Hn+1 (z)) + an+1 • (viin+1 ) For z ∈ Rn ∩ Fn−1 (Rn ) Hn+1 (Fn (z)) = Hn+1 (z) − 1. Let us explain why these properties imply that the action generated by T, f is linearizable. We will apply the criterion of Section 4.1. We can assume c3 ≥ c1 ≥ 1. Define, for n ≥ 0 c3 (n) = c3 (1 + αn + αn αn+1 + αn αn+1 αn+2 + . . . ) < 4c3 . Take c = c2 + 8c3 . We will show that if |Imz| < ∆n −

1 B(αn ) − [c2 + 2c3 (n)] 2π

then for all m ≥ 0, we have | m Fnm (z) − m z| ≤ c3 (n). This allows to conclude by Proposition 4.1 (taking n = 0). Assume by contradiction that for some n ≥ 0, some z and some M > 0, we have 1 B(αn ) − [c2 + 2c3 (n)], 2π m | m Fn (z) − m z| ≤ c3 (n), for 0 ≤ m < M | m z| < ∆n −

| m FnM (z)

− m z| > c3 (n).

(4.1) (4.2) (4.3)

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We may assume that e z ∈ [0, 1). Observe that from (4.1), (4.2) we have 1 B(αn ) − [c2 + 2c3 (n)] 2π 1 log αn−1 − (c2 + 1) < ∆n − 2π

| m Fnm (z)| < ∆n −

for 0 ≤ m < M , hence by (vn ) | m FnM (z)| < ∆n −

1 log αn−1 − c2 . 2π

Thus, if we put km = [e Fnm (z)] for 0 ≤ m ≤ M and wm = Fnm (z) − km , we have wm ∈ Rn for all 0 ≤ m ≤ M . We have also k0 = 0 and km+1 − km ∈ {0, 1, 2} by (vn ). Also, Fn (wm ) = wm+1 + (km+1 − km ). Set um = Hn+1 (wm ) for 0 ≤ m ≤ M . By (vi)n+1 and (vii)n+1 we have k

m+1 um+1 = Fn+1

−km

(um ) + (km+1 − km )an+1 − 1

and thus km um = Fn+1 (u0 ) + km an+1 − m

for 0 ≤ m ≤ M . On the other hand, by (4.3) and (ivn+1 ), we have |αn (m uM − m u0 )| > c3 (n) − c3 and therefore, as c3 (n) = c3 + αn c3 (n + 1), we get | m uM − m u0 | > c3 (n + 1).

(4.4)

We have also by (ivn+1 ): | m u0 | ≤ αn−1 [| m z| + c3 ] 1 B(αn ) − (c2 + 2c3 (n)) + c3 ] < αn−1 [∆n − 2π using (ii) 1 B(αn+1 ) + c3 αn−1 − (c2 + 2c3 (n))αn−1 ≤ ∆n+1 + c1 αn−1 − 2π 1 B(αn+1 ) − (c2 + 2c3 (n + 1)), ≤ ∆n+1 − 2π as c1 ≤ c3 . We conclude that u0 for Fn+1 and some integer M ≤ kM satisfy the same properties (4.1), (4.2), (4.3) of z for Fn and M . But we have, by (vn ) kM < e(FnM (z) − z) + 1
0 such that, if 0 ≤ m z ≤ ∆ −

1 log α0−1 − c8 = h 2π

then 1 α0 , 10 1 α0 , |Df (z) − 1| ≤ 10 1 α0 . |D2 f (z)| ≤ 10

|f (z) − z − α0 | ≤

(4.8) (4.9) (4.10)

Therefore, relation (v) in Section 4.2.1 will follow provided that c2 ≥ c8 . Consider now the Jordan domain U ⊂ B∆ whose boundary is the union of the vertical segment joining ih to −ih, the image f () and the segments joining ih to f (ih), −ih to f (−ih) (see Figure 1). By (4.8), (4.9) the curve we have described has no self-intersection. We glue the two vertical-like sides of U through the holomorphic map f ; we thus obtain an abstract annular ˜ . Complex conjugation leaves U invariant and induces an Riemann surface U ˜ . Define antiholomorphic automorphism of U ∆ =

1 ˜) mod(U 2

˜ is obviously not biholomorphic to C∗ , thus ∆ < +∞). We now choose (U ˜ 1 from U ˜ onto B∆ /Z, which is orientation reversing on a biholomorphism H ˜ onto the equators of these annular domains (sending the top boundary of U ˜ the bottom one of B∆ /Z); we lift H1 to an holomorphic map H1 : U ∪ ∪ f () → B∆ which satisfies H1 (f (z)) = H1 (z) − 1,

(4.11)

for all z ∈ . To extend the domain of H1 we will use the Lemma 4.2 Define R = {| e z| < 2, | m z| < ∆ −

1 log α−1 − c2 } 2π

with c2 = c8 + 1. If z ∈ R and e z ≥ 0, there exists a unique integer L ≤ 0 such that f L (z) ∈ U ∪ and | m f j (z)| < ∆ −

1 log α−1 − c8 2π


151

ih f(w)

w α

0

1

-ih

Fig. 1. The construction of the domain U

for 0 ≥ j ≥ L. If z ∈ R and e z < 0, there exists a unique integer L > 0 such that f L (z) ∈ U ∪ and | m f j (z)| < ∆ −

1 log α−1 − c8 2π

for 0 ≤ j ≤ L. Proof. It is clear from (4.8).

We now extend the domain of definition of H1 to U ∪ R using the previous Lemma and the relation (4.11). By the Lemma, relation (4.11) will hold as soon as z ∈ R ∩ f −1 (R). Next we will define F1 (from property (vi1 ) in Section 4.2.1): set V = H1 ((U ∪∪f ())∩R), V ∗ = ∪n∈Z T n (V ), and denote by ∆∗ the largest ∆1 such that B∆1 ⊂ V ∗ (see Figure 2). For z ∈ (U ∪∪f ())∩R, define F1 (H1 (z)) = H1 (z − 1) − a1

(4.12)

(which is possible as z − 1 ∈ R); if z ∈ ∩ R, we will have F1 (H1 (z) − 1) = F1 (H1 (f (z))) = H1 (f (z) − 1) − a1 = H1 (f (z − 1)) − a1 = H1 (z − 1) − 1 − a1 = F1 (H1 (z)) − 1. This shows that F1 extends from V to V ∗ as an holomorphic map, commuting with T . Also F1 obviously restricts to a diffeomorphism of the real line. One

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l

R ∩U

V

V*

Fig. 2. Standard construction: glueing, uniformizing, developing on the plane

also easily checks that F1 is univalent on V ∗ , and that relation (4.12) extends to all z ∈ R ∩ T −1 (R). We now have constructed F1 and H1 , and we have checked properties (i1 ), (iii1 ), (vi1 ), (vii1 ) of Section 4.2.1. We have also seen how (v) follows from the choice of R. There remains to check (ii1 ) and (iv1 ). Both follow from minorations of moduli of annular domains and Gr¨ otzsch’s Theorem. For −h ≤ y ≤ h, denote by σy the segment joining iy to f (iy). For 0 < y < h, let Ay (resp. A+ y ) be the Jordan domain whose boundary is formed of σ0 , σy , [0, iy] and f ([0, iy]) (resp. σh , σy , [iy, ih] and f ([iy, ih])). We glue through f the vertical sides of Ay , A+ y to obtain annular Riemann . surfaces A˜y , A˜+ y In the Appendix we will show that |αmodA˜y − y| ≤ c9 , |αmodA˜+ y − (h − y)| ≤ c10 .

(4.13) (4.14)


153

1 We obtain (ii1 ) (i.e. a bound from below for ∆∗ ) taking y0 = ∆− 2π log α−1 − c2 − 1. Then, Ay0 ⊂ U ∩ R and therefore

1 mod (V ∗ /Z) ≥ α−1 (y0 − c9 ) 2 otzsch. To obtain (iv1 ) consider z ∈ R with m z > 0. and (ii1 ) follows by Gr¨ It is sufficient to consider z ∈ U ∪ (with m z > 0) because of (vii1 ) and (v0 ): the point z = f L (z) given by the Lemma above satisfies | m z − m z | < 1. On the other hand, such a z lies on a unique segment σy with | m z − y| ≤

1 . 10

From Gr¨ otzsch’s Theorem, we now have a bound from below for | m H1 (z )| because modA˜y ≥ α−1 (y − c9 ) and a bound from above because −1 (h − y − c10 ) modA˜+ y ≥α

which together prove (iv1 ). This concludes the geometric part of the proof of the Theorem. 4.3

Local Theorem 1.3: small strips

The local Theorem 1.3 for small strips is an easy consequence of the local Theorem 1.2 for big strips. Consider f ∈ Diff ω + (R), commuting with T , with rotation number α ∈ B, holomorphic in a strip B∆ . We will now assume that ε(f ) = sup |f (z) − z − α| B∆

is small. Let η ( 1 to be chosen later. We perform a construction similar to the one described in Section 4.2.2: consider the curve made of the segments joining i(∆ − η) to −i(∆ − η), i(∆ − η) to f (i(∆ − η)) and −i(∆ − η) to f (−i(∆ − η)) and the image f ([−i(∆ − η), i(∆ − η)]); this is a Jordan curve if ε(f ) is sufficiently small, namely ε(f ) < ε0 (∆, η, α) (use Cauchy estimates to control f ([−i(∆ − η), i(∆ − η)])).

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Next we glue through f the vertical sides of the Jordan domain U bounded ˜ . We also define by this curve, to get an annular Riemann surface U R = {| e z| < 2, | m z| < ∆ − 2η}. As in Section 4.2.2, we can define a holomorphic map H1 on R, uniformizing ˜ , and a diffeomorphism F1 , commuting with T , and conjugated to T f a1 by U H1 , holomorphic on a strip B∆∗1 . The key fact is the following Lemma 4.3 Given ∆1 < α−1 ∆ and ε1 > 0, we can choose η ( 1 and ε0 such that, if we start with ε(f ) < ε0 , then we obtain F1 holomorphic on B∆1 and ε(F1 ) = sup |F1 (z) − z − α1 | < ε1 . B∆1

The easy proof is omitted. The local Theorem 1.3 now follows from the local Theorem 1.2 by iteration −1 of the Lemma 4.3: for all N > 0, ∆N < βN −1 ∆, εN > 0, there exists ε¯0 = ε¯0 (∆N , εN , α, ∆) such that, starting with ε(f ) < ε¯0 we end up after N renormalization steps with a diffeomorphism FN ∈ Diff ω + (R), commuting with T , holomorphic on B∆N , conjugated to T pN f qN and satisfying ε(FN ) = sup |FN (z) − z − αN | < εN . B∆N

Actually, by Cauchy estimates, we can as well assume that FN is univalent on B∆N . But then, if N is large enough, we can apply local Theorem 1.2 to FN : indeed we have, if

βn−1 log αn−1
0, there exists n and a diffeomorphism k ∈ Diff ω + (R) such that k ◦ fn ◦ k −1 = T k ◦ fn+1 ◦ k −1 = Fn+1 where Fn+1 is holomorphic and univalent on B∆0 , with rotation number αn+1 . be a small parameter to be Proof. We may assume ∆0 > 1. Let η ( ∆−1 0 determined later. By Proposition 3.21, we can find an analytic conjugacy for the action sending (for some m = m(η, ∆0 , f )) fm to T and fm+1 to a diffeomorphism F with sup |D log DF (x)| < η∆−1 0 . R

Therefore, we can as well assume that f = F satisfies this estimate. We then pick ∆ > 0 such that f is holomorphic and univalent on B ∆ with τ := sup |D log DF (z)| < 2η∆−1 0 . B∆

Then ∆0 log 2 1 − > 2τ 2 8η provided that η is small enough. We choose n large enough to have also ∆0 ∆ > 2Mn 8η

and Mn1/2 < η.

Then the estimates in the proposition hold for |w0 | ≤ (8η)−1 ∆0 . In particular, taking η < 1/24 and w0 ≤ 3∆0 , we will have, for j ≤ qn+1 : % % % % %log dwj % ≤ 14η, (4.16) % dw0 % % % % % %log wj % ≤ 14η. (4.17) % w0 % and also, by Corollary 3.20

% % % % %log |In (fn (x))| % < c4 η, % |In (x)| % % % % % %log |In (fn+1 (x))| % < c4 η. % % |In (x)|

(4.18) (4.19)


157

We now perform the same geometric construction as in Section 4.2 or 4.3: set h = 2∆0 |In (0)|; for η ( ∆−1 0 , consider the Jordan domain U whose “vertical” sides are the segment [−ih, ih] and its image by fn , and horizontal sides are [−ih, fn (−ih)], [ih, fn (ih)]; glue the vertical sides of U through f ; uniformize the resulting annular Riemann surface to get an holomorphic map k : U → C which is real and orientation-reversing on U ∩ R, extends continuously to the vertical sides of U , and satisfies k(fn (it)) = k(it) − 1

(4.20)

for |t| < h. Next, defining the rectangle 7 8 3 R = | e z| ≤ 3|In (0)|, | m z| ≤ h , 4 we use (4.16) and (4.18) (with η ( ∆−1 0 ) to extend through relation (4.20) the domain of k to U ∪ R. Setting 8 7 2 U1 = z ∈ U, | m z| < h , 3 we deduce from (4.16), (4.18) that fn+1 (U1 ) ⊂ R and this allows to define a holomorphic map Fn+1 on k(U1 ) by Fn+1 (k(z)) = k(fn+1 (z)), z ∈ U1 . We will be finished if the strip B∆0 is contained in the union ∪m∈Z T m (k(U1 )). But this will be clearly true if η is small enough. 4.5

Global Theorem: proof of linearization

4.5.1 The geometric construction with large strip but small rotation number Let π : Z2 → Diff ω + (R) be an analytic action with π(e−1 ) = T , π(e0 ) = f , ρ(π) = [−1, α], α ∈ (0, 1). We assume that f is holomorphic and univalent in a strip B∆ , with ∆ ≥ ∆min , ∆min a universal constant to be determined later. We also assume that α is so small that ∆≤

1 log α−1 + c0 , 2π

(4.21)

with c0 another universal constant determined as follows: if on the opposite 1 ∆ > 2π log α−1 + c0 , we are able to perform the construction of Section 4.2.2 with the estimates indicated in Section 4.2.1.

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From relations (4.5), (4.6) and (4.7) we get, for | m z| ≤ 1: |f (z) − z − α| ≤ c5 e−2π∆ ,

(4.22)

c6 e−2π∆ , c7 e−2π∆ ,

(4.23)

|Df (z) − 1| ≤ |D log Df (z)| ≤

(4.24)

and then, joining (4.21) with (4.22) |f (z) − z| ≤ c11 e−2π∆ .

(4.25)

We apply Proposition 4.4 to f in the strip B1 with n = 0. We have 1 log 2 1 |w|max = min − ≥ c12 e2π∆ . , 2M0 2τ 2 Let h = θe2π∆ |f (0)|, with θ a universal constant conveniently small, in particular θ < 12 c12 . If we have also c6 θ ( 1, then we can define a Jordan domain U whose boundary is formed of [−ih, ih], its image under f , and the segments [−ih, f (−ih)], [ih, f (ih)]. We glue the vertical sides of U through f , and uniformize the resulting Riemann surface to get a map H : U → C, real and orientation-reversing on U ∩ R, which extends continuously to the vertical sides of U and satisfies H(f (z)) = H(z) − 1, z ∈ .

(4.26)

For small θ, we have from (4.23) 1 |f (0)| (4.27) 10 if | e z| ≤ 3|f (0)|, | m z| ≤ h. This allows to extend the domain of H, through (4.26), to 8 7 3 R = | e z| ≤ 3|f (0)|, | m z| ≤ h . 4 |f (z) − z − f (0)| ≤

Now, Proposition 4.4 shows that (provided θ is small) if z ∈ U and | m z| ≤ 1 a1 2 h, then T f (z) ∈ R. Therefore we can define F on V = H(U ∩ Bh/2 ) by F (H(z)) = H(T f a1 (z)), z ∈ U ∩ Bh/2 .

(4.28)

Then F will extend to an holomorphic and univalent map on ∪m∈Z T (V ), commuting with T . It remains to estimate the witdth of the largest strip B∆∗ contained in ∪m∈Z T m (V ). By the Grotzsch’s Theorem mentioned earlier, this amounts to obtain a bound from below for the modulus of the annular Riemann surface V˜ obtained by glueing through f the vertical sides of V . In the appendix, we will show that m

modV˜ > c13 θe2π∆ which gives finally (with now a fixed value of θ) ∆∗ ≥ c14 e2π∆ .

(4.29)


159

4.5.2 Conclusion of the proof of the global theorem The proof of the global theorem is now easy from the above estimates and the local theorem. Recall from Section 2.5 the definition (2.11) of the functions rα (α ∈ (0, 1)) involved in the statement of conditon H. On the other side, we define, in connection with 4.2.1 (ii) and (4.29), for α ∈ (0, 1), ∆ ∈ R

1 1 if ∆ ≥ 2π log α−1 − c1 log α−1 + c0 , α−1 ∆ − 2π ∗ (4.30) rα (∆) = 1 c14 e2π∆ if ∆ < 2π log α−1 + c0 , where c0 has been chosen such that c0 > c1 + 1. ∗

Lemma 4.6 For any c > 0, there exists ∆0 such that, for any α ∈ (0, 1), ∗ any ∆∗0 ≥ ∆0 , any k ≥ 0, one has 1 rα ◦ · · · ◦ rα0 (0). 2π k−1 Proof. Left as an exercise to the reader. rα∗ k−1 ◦ · · · ◦ rα∗ 0 (∆∗0 ) ≥ c +

Let now α ∈ H and π : Z2 → Diff ω + (R) be an action with π(e−1 ) = T , π(e0 ) = f , ρ(π) = [−1, α]. With the constant c involved in the statement of the local theorem (large strips), we apply the Lemma above which gives a ∗ constant ∆0 . We then make use of Corollary 4.5: we find N and k ∈ Diff ω + (R) such that k ◦ fN −1 ◦ k −1 = T, k ◦ fN ◦ k −1 = F, with F holomorphic and univalent in B∆∗ and ρ(F ) = αN . For k > 0, we 0 now set ∗

∆∗k = rα∗ N +k−1 ◦ · · · ◦ rα∗ N (∆0 ), xk = rαN +k−1 ◦ · · · ◦ rαN (0). By the very definition of condition H, there exists K such that xK ≥ B(αN +K ) which implies, by the Lemma above 1 B(αN +K ). 2π On the other side, starting from F , the geometric constructions of Sections 4.2.2 or 4.5.1 allow to obtain a sequence of diffeomorphisms Fk , 0 ≤ k ≤ K such that ∆∗K ≥ c +

• the action generated by T, Fk is analytically conjugated to that generated by T, F ; • Fk is holomorphic and univalent in B∆∗k , with rotation number αN +k . But ∆∗K is large enough to apply the local theorem: the action generated by T, FK (and thus also the initial action) is analytically linearizable.

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4.6

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Global Theorem: Construction of nonlinearizable diffeomorphisms

4.6.1 Introduction In this last section, we will construct, for any α ∈ / H, analytic actions with rotation number [−1, α] which are not linearizable. Actually we get even a slightly more precise result, relating the way in which condition H fails to the width of the strips of univalence for the examples. The construction of nonlinearizable actions is based on a geometric construction analogous to, but more sophisticated than, those which were made in Section 4.2.2. It is explained in Section 4.6.4. In the next section, we will state the result of this geometric construction; in Section 4.6.3, we explain how to derive the construction of analytic nonlinearizable actions from the basic step 4.6.2. 4.6.2 Result of the geometric construction In this section, we are given an analytic action π : Z2 → Diff ω + (R) with π(e−1 ) = T , π(e0 ) = f and ρ(π) = [−1, α], with α > 1. We assume that f is holomorphic and univalent in a strip B∆ . Proposition 4.7 There exist universal constants c15 , c16 , . . . , c20 such that, if ∆ > c15 we are able to construct an analytic diffeomorphism F , commuting with T , with rotation number α−1 and the following properties: (i)

F is holomorphic and univalent in a strip B∆ with

1 log α − c17 if α < c16 ∆ α−1 ∆ + 2π ∆ = 1 if α ≥ c16 ∆. 2π log ∆ − c18

(ii) If α ≥ c16 ∆, F has a fixed point p with 0 < m p < c19 < ∆ . (iii) Assume that, for some P, Q > 0, f has a periodic orbit (modZ) O with f Q (z) − P = z, z ∈ O, 0 < max m z < 2c19 ; O

then F has a periodic orbit O with f P (z ) − Q = z , z ∈ O , 0 < max m z < λ max m z, O

where

λ=

4 −1 3α c20 ∆−1

O

if α < c16 ∆ ifα ≥ c16 ∆.

We defer the proof of this proposition to Section 4.6.4.


161

4.6.3 Construction of nonlinearizable actions Before we construct, using Proposition 4.7 as a building block, nonlinearizable analytic diffeomorphisms, we need a few preliminary arithmetical remarks. For α > 1, ∆ > 0, we define, in connection to Section 4.6.2:

1 log α − c17 if ∆ > c−1 α−1 ∆ + 2π 16 α, ∗ Rα (∆) = 1 (4.31) if ∆ ≤ c−1 16 α. 2π log ∆ − c18 This is closely connected to the function (rα−1 )−1 :

α−1 x + log x + 1 if x ≥ α, (rα−1 )−1 (x) = log x if x ≤ α.

(4.32)

∗ The function Rα appears in the third of the following elementary lemmas.

Lemma 4.8 Let α be irrational and x0 ≥ 0. Assume that for all k ≥ 0 we have xk := rαk−1 ◦ · · · ◦ rα0 (x0 ) ≥ log αk−1 . Then α ∈ B and xk ≥ B(αk ) − 4 for all k ≥ 0. / H if and only if there Lemma 4.9 Let α be irrational and x∗ ≥ 0. Then α ∈ exists m and an infinite set I = I(m, x∗ , α) such that, for all k ∈ I −1 . rαm+k−1 ◦ · · · ◦ rαm (x∗ ) < log αm+k

Lemma 4.10 There exists a universal constant c21 > 0 with the following property. Let α be irrational, x0 > 2πc21 , N > 0; define xk := rαk−1 ◦ · · · ◦ rα0 (x0 ), choose ∆N ≥

1 2π xN

+ c21 −

1 2π x0

and define

∗ ∗ ∆k := Rα −1 ◦ · · · ◦ R −1 (∆N ), α k

0 ≤ k ≤ N,

N −1

0 ≤ k ≤ N.

Then, for all 0 ≤ k ≤ N , we have ∆k ≥

1 1 xk − c21 − max 0, xN − ∆N . 2π 2π

All the three lemmas above are elementary computations that are left to the reader. We now select some irrational number α which does not belong to H, and some width ∆∗ ≥ 1. We set 1 log+ c16 , 2π x∗ = 2π(∆∗ + c22 ),

c22 = c18 + c21 +

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J.-C. Yoccoz

and we fix some integer m given by Lemma 4.9 such that the set I = I(m, x∗ , α) in this lemma is infinite. We will construct nonlinearizable analytic diffeomorphisms with rotation number αm which are holomorphic and univalent in B∆∗ . (k) Let k ∈ I. We define diffeomorphisms fl , 0 < l ≤ k + 1, with rotation (k) −1 number αm+l−1 and diffeomorphisms Fl , 0 ≤ l ≤ k, with rotation number αm+l as follows: (k)

• fk+1 = Rα−1 ; m+k

(k)

• Fl

(k)

• fl

(k)

is obtained from fl+1 by Proposition 4.7;

= T −am+l ◦ Fl

(k)

. (k)

We choose to consider fk+1 = Rα−1

m+k

(k)

on the strip B∆(k) with ∆k+1 =

−1 c−1 16 αm+k . It then follows from Proposition 4.7 that Fl morphic and univalent on a strip B∆(k) with

(k)

k+1

(k)

and fl

are holo-

l

(k) ∆l

∗ ∗ = Rα ◦ · · · ◦ Rα −1 −1 (∆k+1 ). (k)

m+l

m+k

Let xl = rαm+l−1 ◦ · · · ◦ rαm (x∗ ). Observe that we have, as k ∈ I: (k)

∆k =

1 1 log αk−1 − c22 + c21 ≥ xk + c21 − c22 2π 2π

and therefore, by Lemma 4.10 (k)

∆l

≥

1 xl − c22 . 2π

In particular, taking l = 0, we have ∆0 ≥ ∆∗ . (k)

On the other hand, by (ii) in Proposition 4.7, for k ∈ I, k ≤ k, Fk has a (k) fixed point pk such that (k)

(k)

0 < m pk < c19 . We take ∆∗ > 2c20 and iterate (iii) in Proposition 4.7; we obtain that F0 (k) has a periodic orbit Ok (modZ) of fixed period Qk with

(k)

0 < max m z < 2 (k)

Ok

9 10

k c19 .

Analytic linearization of circle diffeomorphisms (k)

163

(∞)

Finally, we extract from (F0 )k∈I a subsequence converging to F0 : this is an analytic diffeomorphism, holomorphic and univalent in B∆∗ , with rotation (∞) number αm , which has, for each k ∈ I, a periodic orbit Ok (modZ) of period Qk satisfying 0 < max m z ≤ 2 (∞)

Ok

(∞)

Therefore F0

9 10

k c19 .

is not linearizable. (k)

(k)

Remark 4.11 To obtain pk from (ii) in Proposition 4.7, we need ∆k +1 ≤ −1 c−1 16 αm+k ; this is obtained by definition for k = k; when k < k we can force (k)

this inequality by taking ∆k +1 smaller if necessary. 4.6.4 Proof of the Proposition 4.7 Let f, α, ∆ be as in Proposition 4.7, with ∆ > c15 , c15 large enough. By the estimates at the beginning of Section 4.2.2, there exists c23 such that, for 0 ≤ m z ≤ ∆ − c23 , we have 1 exp[−2π(∆ − c23 − m z)], 10 1 exp[−2π(∆ − c23 − m z)], |Df (z) − 1| ≤ 10 1 exp[−2π(∆ − c23 − m z)]. |D2 f (z)| ≤ 10

|f (z) − z − α| ≤

On the other hand, we choose a Z-periodic function ψ in {m z ≥ ∆−c23 +1} which satisfies there 1 exp[−2π(m z − ∆ + c23 − 1)], 10 1 exp[−2π(m z − ∆ + c23 − 1)], |Dψ(z)| ≤ 10 1 exp[−2π(m z − ∆ + c23 − 1)]. |D2 ψ(z)| ≤ 10 |ψ(z)| ≤

We also choose α ∈ C, with |α − α | ≤ g(z) = z + α + ψ(z),

1 10

and set

m z ≥ ∆ − c23 + 1.

We define also g(z) = g(¯ z ),

m z ≤ −(∆ − c23 + 1).

The most obvious and easiest choices are ψ ≡ 0, α = α, but other choices are allowed and may give interesting properties.

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We define a vertical strip U as follows. The left boundary is the imaginary axis, that we view as the union of the segments [−i(∆ − c23 ), i(∆ − c23 )] = , [i(∆−c23 ), i(∆−c23 +1)] = ∗ , [i(∆−c23 +1), +i∞) = + and the symmetric ¯∗ and ¯+ . The right boundary is the union of the curves f (), g(+ ), g(¯+ ), and the two (symmetric) straight segments ∗1 , ¯∗1 which connect f (i(∆ − c23 )) to g(i(∆ − c23 + 1)) and f (−i(∆ − c23 )) to g¯(−i(∆ − c23 + 1)). The estimates on f , ψ above guarantee that the domain U is well defined. We will also have to consider the square S = {0 ≤ e z ≤ 1, ∆ − c23 ≤ m z ≤ ∆ − c23 + 1}

and its symmetric S. Next, we glue together some parts of the boundary of U : we glue to f () through f , + to g(+ ) through g and ¯+ to g¯(¯+ ) through g¯. We obtain a ˜ which is a twice-punctured annulus, the two punctures Riemann surface U ˜ to ∗ ∪ ∗ , corresponding to +i∞ and −i∞ and the two components of ∂ U 1 ∗ ∗ ¯ ¯ ∪ 1 (see Figures 3 and 4). The fact that ±i∞ give rise to punctures (and not holes) is checked via moduli estimates from the appendix.

g

1

S f

∆ -c 23

α

S

Fig. 3.


165

i

R

R -i

Fig. 4.

˜1 the trace in U ˜ of U \ (S ∪ S). With z ∈ U \ (S ∪ S), the Denote by U formulas z→ z − 1 if e z ≥ 1, z→ f (z − 1) if 0 ≤ e z ≤ 1, | m z| < ∆ − c23 , z → g(z − 1) if 0 ≤ e z ≤ 1, m z > ∆ − c23 + 1, z → g¯(z − 1)

if

0 ≤ e z ≤ 1, m z < −(∆ − c23 + 1),

˜1 → U ˜ which is holomorphic and univalent. induce after glueing a map F˜ : U ˜ Moreover, F extends holomorphically at the punctures and leave these punc˜˜ (resp. U ˜˜ ) the Riemann surface U ˜ (resp. U ˜1 ) tures fixed. We denote by U 1 with the punctures filled in. Let ˜ = modU ˜, 2∆ ˜1 , ˜ 1 = modU 2∆ ˜ :U ˜ → B ˜ be a biholomorphism which is orientation-reversing and let H ∆/Z ˜ Define on the equators. Let H : U \ (∗ ∪ ∗1 ∪ ¯∗ ∪ ¯∗1 ) → C a lift of H. V = H(U \ (S ∪ S)), V ∗ = ∪n∈Z T n (V ). ˜ U ˜1 ). By Grotzsch’s theorem, setting ∆ = ∆ ˜ 1 − 1 log 4, We have V ∗ /Z = H( 2π we have B∆ ⊂ V ∗ .

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On the other side, we define an analytic diffeomorphism F , holomorphic and univalent on B∆ , commuting with T by the condition H(z − 1) = F (H(z)), for z ∈ U , e z > 1. We will have ρ(F ) = α−1 and we need to check the other properties of Section 4.6.2. ˜ 1 from below. Let To get the estimates 4.6.2 (i), we need to bound ∆ ω = i(∆ − c23 + 1/2); consider the sets, for r > 2 AL (r) = {2 < |z − ω| < r} ∩ U , AR (r) = {2 < |z − (ω + α)| < r} ∩ U . If ∆−c23 < α2 , the sets AL (∆−c23 ), AR (∆−c23 ) do not intersect and the trace ˜ is an annular domain A1 contained in U ˜1 and separating of their union in U ˜ the equator of U1 from one of the boundary components. We therefore have ˜ 1 ≥ modA1 ∆ and we will show in the appendix that modA1 ≥

1 log ∆ − c18 2π

provided that ∆ is large enough (∆ > c15 ). If ∆ − c23 ≥ α2 , the sets AL (α/2), AR (α/2) do not intersect and the trace ˜ is an annular domain A2 contained in U ˜1 and separating of their union in U ˜ the equator of U1 from one of the boundary components. We will show in the appendix that we have modA2 ≥

1 log α − c17 . 2π

˜1 When moreover ∆ − c23 ≥ α, we also consider the annular domain A2 ⊂ U whose trace on U is the domain between the segments [0, f (0)] and [ω , f (ω 1 1 )], with ω1 = i ∆ − c23 − α2 . This annular domain is disjoint from A2 and ˜1 from the same boundary component (associated separates the equator of U to S) than A2 . In the appendix, we will show that modA2 ≥

∆ − c17 . α

We will therefore have ˜ 1 ≥ 1 log α − c17 , ∆ 2π


167

if ∆ − c23 ≥ α/2, and even ˜ 1 ≥ 1 log α + α−1 ∆ − c − c , ∆ 17 17 2π when ∆ − c23 ≥ α. ˜ 1 give (i) in 4.6.2. We turn now to (ii). The two The three estimates for ∆ symmetric fixed points we are interested in (when c16 ∆ ≤ α) are H(±i∞) = ˜ : in U ˜ we take out {p− , p+ }, p∓ . We define the following annular domain in U the trace of the line {e z = α/2} and the trace of the segment [−iω, iω]; this is an annular domain A3 which separates {p+ , p− } from the boundary ˜ . In the appendix we will show that components of U modA3 ≥

1 log(α∆−1 ) − c19 ≥ c19 > 0. 2π

˜ of U ˜ (filled-in at ±i∞), we obtain (ii) from Considering the uniformization H standard Teichm¨ uller estimates (see [Ah]). Finally, we consider property (iii) of 4.6.2. We will assume c15 > 2c19 + c23 + 1. Let O be a periodic orbit as in 4.6.2 (iii); let z0 ∈ O with 0 ≤ e z0 < 1. We define inductively zi , i ≥ 0, by zi+1 = zi − 1 if e zi ≥ 0, zi+1 = f (zi ) if e zi < 0. Then we have zP +Q = f Q (z0 ) − P = z0 . On the other hand, let zi = H(zi ); we have zi+1 = zi − 1 if zi+1 = f (zi ), zi+1 = F (zi ) if zi+1 = zi − 1,

hence zP +Q = f P (z0 ) − Q = z0 . Because H is orientation reversing on the equators, the periodic orbit O in 4.6.2 (iii) will actually be the orbit of z¯0 . To estimate m zi , 0 ≤ i < P + Q, we consider H in the ball B of center xi = e zi and radius 12 (∆ − c23 ); the functional equation H(f (z)) = H(z)−1 allows to define H in this ball and to check that it is univalent there. In 4.6.2, we take c15 large enough and c16 small enough. If ∆ ≤ c−1 16 α, the image by H of the interval B ∩ R of length ∆ − c23 by the functional equation will be of length at most 2c−1 16 ; by Koebe’s 1/4-theorem, we will have DH(xi ) ≤ c20 ∆−1 , and by the standard distorsion estimates we will obtain | m zi | < c20 ∆−1 m zi . If ∆ > c−1 16 α, we use H(f (xi )) = H(xi ) − 1 and the standard distorsion estimates to obtain |DH(xi )| ≤ 54 α−1 (with c16 small enough), and then (with c15 large enough) | m zi | < 43 α−1 m zi . This concludes the proof of Proposition 4.6.2 and of the construction of non linearizable analytic diffeomorphisms.

168

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J.-C. Yoccoz

Appendix: Estimates of moduli of annular domains

5.1

Dirichlet integrals

Let A be an annular Riemann surface with boundary ∂A = ∂0 A ∂1 A. Let E(A) be the space of functions H ∈ W 1,2 (A) such that H|∂0 A ≡ 0, H|∂1 A ≡ 1. For H ∈ E(A) define the Dirichlet integral D(H) = |∇H|2 . A

Then, we have (modA)−1 = inf D(H). E(A)

In particular, for any H0 ∈ E(A) modA ≥ D(H0 )−1 . The annular Riemann surfaces for which we need to bound from below the modulus were obtained in two similar but slightly distincts ways. 5.2

First kind of moduli estimates

In the first setting, we have an holomorphic map f (z) = z + ϕ(z). A domain U has for boundary a vertical segment = [ih0 , ih1 ], its image f () and the segments [ih0 , f (ih0 )], [ih1 , f (ih1 )]. We obtain an annular Riemann surface A when we glue and f () through f . For t ∈ [0, 1], h ∈ [h0 , h1 ], set x + iy = ih + tϕ(ih), ϕ1 (h) = e ϕ(ih), ϕ2 (h) = m ϕ(ih). One has dx = ϕ1 (h)dt + tDϕ1 (h)dh dy = ϕ2 (h)dt + (1 + tDϕ2 (h))dh, hence dh = [ϕ1 (h)dy − ϕ2 (h)dx][ϕ1 + t(ϕ1 Dϕ2 − ϕ2 Dϕ1 )]−1 , dx ∧ dy = [ϕ1 + t(ϕ1 Dϕ2 − ϕ2 Dϕ1 )]dt ∧ dh. Set now H(x, y) =

h − h0 . h1 − h0


169

We will have H ∈ E(A) and D(H) = (h1 − h0 )−2

|∇h|2 (5.1) = (h1 − h0 )−2 (ϕ21 + ϕ22 )[ϕ1 + t(ϕ1 Dϕ2 − ϕ2 Dϕ1 )]−1 dtdh (5.2) −2

≤ (h1 − h0 )

h1

h0

5.2.1

|ϕ(ih)|2 dh . ϕ1 (h) − |ϕ1 (h)Dϕ2 (h)| − |ϕ2 (h)Dϕ1 (h)|

(5.3)

˜y , A ˜+ in 4.2.2 We have there, with Estimates for A y hmax = ∆ −

1 log α0−1 − c8 2π

the following estimates for 0 ≤ h ≤ hmax : |ϕ(ih) − α0 | ≤ c5 α0 exp(−2πc8 ) exp[−2π(hmax − h)], |Dϕ(ih)| ≤ c6 α0 exp(−2πc8 ) exp[−2π(hmax − h)]. Plugging in these estimates in (5.1), we obtain D(H) ≤ (h1 − h0 )−1 α0 + c9 (h1 − h0 )−2 α0 and then (D(H))−1 ≥ α0−1 (h1 − h0 )[1 − c9 (h1 − h0 )−1 ] which gives the required estimates for the moduli of A˜y , A˜+ y. 5.2.2 Estimate for V˜ in 4.5.1 One takes h0 = 0, h1 = 13 θ exp(2π∆)|f (0)|; one has |f (z) − z − f (0)| ≤

1 |f (0)|, 10

(4.27)

for | e z| ≤ 3|f (0)|, | m z| ≤ h, and |Df (z) − 1| ≤ c6 exp(−2π∆)

(4.23)

for | m z| ≤ 1. Plugging in these estimates in (5.1) gives the required estimate: modV˜ ≥ c13 θ exp(2π∆).

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˜ in 4.6.4 The glueing there is 5.2.3 Estimates for the punctures of U made through g(z) = z + α + ψ(z), for z = ih, h ≥ ∆ − c23 + 1, with 1 exp[−2π(m z − ∆ + c23 − 1)], 10 1 exp[−2π(m z − ∆ + c23 − 1)]. |Dψ(z)| ≤ 10 |ψ(z)| ≤

From (5.1), we immediately get that the corresponding annulus has infinite modulus. 5.2.4 Estimate for the modulus of A2 in 4.6.4 The glueing is made through f , which satisfies 1 exp[−2π(∆ − c23 − m z)], 10 1 exp[−2π(∆ − c23 − m z)], |Df (z) − 1| ≤ 10

|f (z) − z − α| ≤

with h0 = 0 and h1 = ∆ − c23 − α2 . It is assumed that ∆ > c15 , ∆ − c23 ≥ α. We obtain in (5.1): D(H) ≤ (h1 − h0 )−2 [α(h1 − h0 ) + c˜17 ] which gives the required estimate for A2 : modA2 ≥ ∆α−1 − c17 . 5.3

Second kind of moduli estimates

The second setting is the following. On one side, we have an exactly semicircular domain AL = {r0 < |z − ω| < r1 , e(z − ω) > 0} and on the other an approximately semicircular domain AR , whose boundary is made from • the two arcs f ± ([ω ± ir0 , ω ± ir1 ]); • the two left half-circles with diameters [f − (ω − irj ), f + (ω + irj )], j = 0, 1; the maps f − , f + here are close to a translation by some real number α > 1. We glue the “vertical” sides of AL , AR through f + , f − to obtain A. For r0 ≤ r ≤ r1 , 0 ≤ t ≤ 12 , we set z(t, r) = x(t, r) + iy(t, r) = ω +

1 exp(2πit)r. i


171

On the other hand, for 1/2 ≤ t ≤ 1, r0 ≤ r ≤ r1 , we set 1 z(t, r) = x(t, r) + iy(t, r) = fc (r) + ρ(r) exp(2πit), i with 1 − [f (ω − ir) + f + (ω + ir)] 2 1 ρ(r) = [f + (ω + ir) − f − (ω − ir)]. 2i

fc (r) =

The formula log r/r0 log r1 /r0

H(z(t, r)) = defines a function in E(A), with

−2 r1 D(H) = log |∇(log r)|2 r0 A r1 r1 dr 2 = π log . |∇(log r)| = π r r 0 AL r0 In AR , we write dx + idy = adr + bdt with 1 a = a0 + ia1 = Dfc (r) + Dρ(r) exp(2πit), i b = b0 + ib1 = 2πρ(r) exp(2πit), giving

1

|∇(log r)|2 = AR

1/2

r

r0

|b|2 dtdr. |a0 b1 − a1 b0 |

Now, we have 5−1 4 |Dfc | Dρ |b|2 − ≤ 2π e |a0 b1 − a1 b0 | ρ ρ giving

|∇(log r)| ≤ π 2

AR

r1

r0

4

e

Dρ ρ

|Dfc | − ρ

5−1 dr.

(5.4)

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5.3.1 Estimates for A1 , A2 in 4.6.4 We take f + = g, f − = f , r0 = 3, ω = i ∆ − c23 + 12 and

∆ − c23 − 1 if 5 ≤ ∆ − c23 < α2 r1 = α if ∆ − c23 ≥ α2 ≥ 5. 2 −1 We have then, from the estimates on f , g in 4.6.4 % % 4 % % %ρ(r) − α − α + r % ≤ 1 exp −2π r − % % 10 2i 4 1 |Dρ(r) − 1| ≤ exp −2π r − 10 4 1 |Dfc (r)| ≤ exp −2π r − 10

5 1 , 2 5 1 , 2 5 1 . 2

When we plug these estimates in (5.4), we obtain r1 |∇(log r)|2 ≤ π log + c18 r0 AR giving the required estimates for modA1 , modA2 in 4.6.4. 5.3.2 Estimate for A3 in 4.6.4 We take now (with c16 ∆ ≤ α, c16 small) ω = 0, f + = g, f − = g¯, r0 = ∆ − c23 + 2, r1 = α2 − 1. The same computation as above will give r1 |∇(log r)|2 ≤ π log + c18 r 0 AR and the required estimate for modA3 . Acknowledgements. I am extremely grateful to Timoteo Carletti, who wrote

a first version of these notes, which served as a basis to the present manuscript. I would also like to thank very much Stefano Marmi, whose friendly pressure and constant help finally allowed this text to exist.

References [Ah]

L.V. Ahlfors: Lectures on Quasiconformal Mappings, Van Nostrand (1966) [Ar1] V.I. Arnol’d: Small denominators I. On the mapping of a circle into itself. Izv. Akad. Nauk. Math. Series 25, 21–86 (1961) [Transl. A.M.S. Serie 2, 46, (1965)] [DMVS] W. De Melo and S.J. Van Strien: One-dimensional dynamics, SpringerVerlag, New York, Heidelberg, Berlin (1993)

Analytic linearization of circle diffeomorphisms [De]

173

A. Denjoy: Sur les courbes définies par les équations differentielles ` a la surface du tore, J. Math. Pures et Appl. 11, série 9, 333–375, (1932) [HW] G.H. Hardy and E.M. Wright: An introduction to the theory of numbers, 5th edition Oxford Univ. Press [He1] M.R. Herman: Sur la conjugaison differentielle des difféomorphismes du cercle a ` des rotations. Publ.Math. I.H.E.S. 49, 5–234 (1979) [He2] M.R. Herman: Simple proofs of local conjugacy theorems for diffeomorphisms of the circle with almost every rotation number, Bol. Soc. Bras. Mat. 16, 45–83 (1985) [KO1] Y. Katznelson and D. Ornstein: The differentiability of the conjugation of certain diffeomorphisms of the circle, Ergod. Th. and Dyn. Sys. 9, 643–680 (1989) [KO2] Y.Katznelson and D.Ornstein: The absolute continuity of the conjugation of certain diffeomorphisms of the circle, Ergod. Th. and Dyn. Sys. 9, 681– 690 (1989) [KS] K.M. Khanin and Ya. Sinai: A new proof of M. Herman’s theorem, Commun. Math. Phys. 112, 89–101 (1987) [MMY1] S. Marmi, P. Moussa and J.-C. Yoccoz: The Brjuno functions and their regularity properties, Communications in Mathematical Physics 186, 265–293 (1997) [MMY2] S. Marmi, P. Moussa and J.-C. Yoccoz: Complex Brjuno functions, Journal of the American Mathematical Society, 14, 783–841 (2001) [PM] R. Pérez-Marco: Fixed points and circle maps, Acta Math. 179, 243–294 (1997) [Ri] E. Risler: Linéarisation des perturbations holomorphes des rotations et applications, Mémoires S.M.F. 77 (1999) [SK] Ya.G. Sinai and K.M. Khanin: Smoothness of conjugacies of diffeomorphisms of the circle with rotations, Russ. Math. Surv. 44, 69–99 (1989) [Yo1] J.-C. Yoccoz: Conjugaison differentielle des difféomorphismes du cercle dont le nombre de rotation vérifie une condition Diophantienne, Ann. Sci. Ec. Norm. Sup. 17, 333–361(1984) [Yo2] J.-C. Yoccoz: Théorème de Siegel, polynˆ omes quadratiques et nombres de Brjuno, Astérisque 231, 3–88 (1995)

Some open problems related to small divisors S. Marmi, J.-C. Yoccoz

0

Introduction

What follows is a redaction of a (memorable) three–hours–long open problem session which took place during the workshop in Cetraro. Each of the five lecturers (H. Eliasson, M. Herman, S. Kuksin, J.N. Mather and J.-C. Yoccoz) spent about half an hour briefly introducing some open problems. This redaction grew from the notes that the first author took during the session: whereas we mention who suggested each of the problems of the list we give below (with the exception of the authors of this text) we are the only responsible for any mistake the reader may find in their description or formulation. Moreover the list of references is very far from being complete and has not been updated. We tried to make this text self-contained, but see [KH] for terminology and further information and [Yo2] for a short survey of classical results concerning small divisor problems.

1 1.1

One-Dimensional Small Divisor Problems (On Holomorphic Germs and Circle Diffeomorphisms) Linearization of the quadratic polynomial. Size of Siegel disks

Let us consider the linearization problem for the quadratic polynomial Pλ (z) = λ(z − z 2 ) ([Yo3], Chapter II) where z ∈ C, λ = e2πiα and α ∈ C/Z. We say that Pλ is linearizable if there exists a holomorphic map tangent to the identity hλ (z) = z + O (z 2 ) such that hλ (λz) = Pλ (hλ (z)). Then hλ is unique and we will denote rλ its radius of convergence (when |λ| = 1 this measures the “size” of the Siegel disk of Pλ ). The second author proved the following results: (1) there exists a bounded holomorphic function U : D → C such that for all λ ∈ D, |U (λ)| is equal to rλ ; (2) for all λ0 ∈ S1 , |U (λ)| has a non-tangential limit in λ0 , which is still equal to rλ0 ; (3) if λ = e2πiα , α ∈ R \ Q, Pλ is linearizable if and only if α is a Brjuno number: if (pn /qn )n≥0 denotes the sequence of the convergents of the continued ∞ fraction expansion of α then being a Brjuno number means that n=0 logqqnn+1 < +∞;


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(4) There exists a universal constant C1 > 0 and for all ε > 0 there exists Cε > 0 such that for all Brjuno numbers α one has (1 − ε)B(α) − Cε ≤ − log |U (e2πiα )| ≤ B(α) + C1 . Problem 1.1.1 Does the function (defined on the set of Brjuno numbers) α → B(α) + log |U (e2πiα )| belong to L∞ (R/Z)? There is a good numerical evidence [Mar] in support to a positive answer to the following much stronger property: Problem 1.1.2 Does the function α → B(α) + log |U (e2πiα )| extend to a 1/2-H¨ older continuous function as α varies in R? These two problems are discussed to some extent in [MMY1], [MMY2]. For some related analytical and numerical results concerning some areapreserving maps, including the standard family, we refer to [Mar], [BPV], [MS], [Da], [BG], [CL]. 1.2

Herman rings. Differentiable conjugacy of diffeomorphisms of the circle

The second author [Yo5] proved that the Brjuno condition is also necessary and sufficient in the local conjugacy problem for analytic diffeomorphisms of the circle. In this case the simplest non-trivial model is provided by the z+a Blaschke products Qa,α (z) = ρα z 2 1+az . Here a ∈ (3, +∞) and ρα ∈ S1 is chosen in such a way that the rotation number of the restriction of Qa,α to S1 is exactly α. Under these assumptions Qa,α induces an orientationpreserving analytic diffeomorphism of S1 . Note that when a → +∞ then Qa,α (z) → e2πiα z. When α is a Brjuno number if a is large enough then Qa,α is analytically conjugated to the rotation Rα (z) = e2πiα z in a neighborhood of S1 . If α satisfies the more restrictive artihmetical condition H (we refer to Yoccoz’s lectures in this volume for its definition) then Qa,α is conjugated to the rotation for all a > 3. This leads to the following Problem 1.2.1 Let α be a Brjuno number not satisfying condition H: does there exist an a > 3 such that Qa,α is not analytically conjugated to the rotation Rα ? Concerning this problem Herman showed that there exists at least a Brjuno number α not satisfying H such that the answer to the previous question is positive. In general one expects to exist a maximal interval (a0 , +∞), a0 > 3, such that Qa,α is analytically conjugated to a rotation for all a ∈ (a0 , +∞) whereas Qa0 ,α is not analytically conjugated. Problem 1.2.2 How smooth is the conjugacy for a0 ? How does this smoothness depend on α?


177

This leads naturally to the study of conjugacy classes of orientationpreserving diffeomorphisms of the circle with finitely many continuous derivatives. Here the relevant arithmetical conditions on the rotation number are of Diophantine type. Let τ ≥ 0; we denote DC(τ ) the set of irrational numbers α whose denominators qn of the continued fraction expansion satisfy qn+1 = O(qn1+τ ) for all n ≥ 0. Let T = R/Z; r, s ∈ {0, +∞, ω} ∪ [1, +∞). Let Diffr+ (T) be the group of r C diffeomorphisms1 of T which are orientation-preserving. We denote Dr (T) the group of C r diffeomorphisms of the real line such that f˜−id is Z-periodic. We consider the linearization problem f ◦ h = h ◦ Rα where Rα denotes the rotation of α on T, α is the rotation number of f (mod 1) and h ∈ Diffs+ (T), with r ≥ s. One must distinguish the local conjugacy problem from the global one: thus we define Cr,s = {α ∈ R \ Q , every f ∈ Diffr+ (T) with rotation number α mod 1 is conjugated to Rα with a conjugacy h ∈ Diffs+ (T)} loc Cr,s = {α ∈ R \ Q , every f ∈ Diffr+ (T) with rotation number α mod 1

C r -close to Rα is conjugated to Rα with a conjugacy h ∈ Diffs+ (T)} loc the neighborhood in the C r topology which Note that in the definition of Cr,s measures the distance of f from Rα depends on α. Let s < r − 1 < ∞. The following inclusions are known after [He2, KO1, KO2, KS, Yo1, Yo4], etc. loc DC(r − s − 1 − ε) ⊂ Cr,s ⊂ Cr,s ⊂ DC(r − s − 1 + ε) for all ε > 0

The third inclusion is due to Herman [He2]. One also knows from [SK] that: • if 1 < s < 2 < s + 1 < r < 3 then DC(r − s − 1) ⊂ Cr,s ; • DC(0) ⊂ Cr,r−1 provided that r > 2, r ∈ / N. loc Problem 1.2.3 Determine Cr,s and Cr,s . Are they different ?

1.3

Gevrey classes

In the case of the conjugacy of germs of formal diffeomorphisms of (C, 0) one can consider a problem similar to the local one above requiring the formal germs to belong to some ultradifferentiable class, for example Gevrey classes. Consider two subalgebras A1 ⊂ A2 of zC[[z]] closed with respect to the composition of formal series. In addition to the usual cases zC[[z]] (formal germs) and zC{z} (analytic germs) one can consider Gevrey for example n s classes Gs , s > 0 (i.e. series F (z) = such that there exist n≥0 fn z 1

If r = 0 it is the group of homeomorphisms of T; if r ≥ 1, r ∈ R \ N, it is the group of C [r] diffeomorphisms whose [r]-th derivative satisfies a H¨ older condition of exponent r − [r]; if r = ω it is the group of R-analytic diffeomorphisms.

178


c1 , c2 > 0 such that |fn | ≤ c1 cn2 (n!)s for all n ≥ 0). Let F ∈ A1 being such that F (0) = λ ∈ C∗ . We say that F is linearizable in A2 if there exists H ∈ A2 tangent to the identity and such that F ◦ H = H ◦ Rλ where Rλ (z) = λz. Let λ = e2πiα with α ∈ R \ Q. One knows that if α is a Brjuno number then for all s > 0 all germs F ∈ Gs have a linearization H ∈ Gs (see [CM]). Let r > s > 0 and denote Gr,s = {α ∈ R\Q , every F ∈ Gs is conjugated to Rα with a conjugacyH ∈ Gr } One knows that a condition weaker than Brjuno is sufficient [CM]. Problem 1.3.1 Determine Gr,s . Of course one can ask a similar question in the circle case, distinguishing the local from the global case.

2 2.1

Finite-Dimensional Small Divisor Problems Linearization of germs of holomorphic diffeomorphisms of (Cn , 0)

Let n ≥ 2 and let f ∈ (C[[z1 , . . . zn ]])n be a germ of formal diffeomorphism of (Cn , 0), z = (z1 , . . . , zn ), f (z) = Az + O(z 2 ) with A ∈ GL (n, C). Let λ1 , . . . , λn denote eigenvalues of A, k = (k1 , . . . , kn ) ∈ Nn , λk = the n k1 kn λ1 . . . λn and |k| = j=1 |kj |. Assume that the eigenvalues are all distinct. If λk − λj = 0 for all j = 1, . . . , n and k ∈ Nn , |k| ≥ 2 (NR) then f is formally linearizable, i.e. there exists a unique germ h of formal diffeomorphism of (Cn , 0), tangent to the identity, such that h−1 ◦ f ◦ h = A. Let f be C-analytic and assume that A satisfies (NR). For m ∈ N, m ≥ 2 let Ω(m) = inf |λk − λj | . 2≤|k|≤m 1≤j≤n

Then Brjuno [Br] proved that if A is diagonalizable, satisfies (NR) and the condition ∞ 2−k log(Ω(2k+1 ))−1 < +∞ (B) k=0

then f is analytically linearizable, i.e. the formal germ h defines a germ of C-analytic diffeomorphism of (Cn , 0). The proof uses the classical majorant series method used by Siegel [S, St] to prove that h is C-analytic under the stronger assumption that λ1 , . . . , λn satisfy a diophantine condition. Problem 2.1.1 (M. Herman) What is the optimal arithmetical condition on the eigenvalues of the linear part which assures that f is analytically linearizable? Can one obtain it by direct majorant series method?


179

It seems unlikely that condition (B) is optimal. Concerning the problem of linearization of germs of holomorphic diffeomorphisms near a fixed point [He4] contains many other questions, most of which are still open. 2.2

Elliptic fixed points and KAM theory

If one replaces the assumption of being conformal with the assumption of preserving the standard symplectic structure of R2n one can consider the following problem. Let f be a real analytic symplectic diffeomorphism of R2n which leaves the origin fixed f (0) = 0. Let z ∈ R2n and assume that f (z) = Az + O(z 2 ), where A ∈ Sp (2n, R) is conjugated in Sp (2n, R) to rα1 × . . . × rαn , where cos 2παi sin 2παi and the vector α = (α1 , . . . , αn ) ∈ Rn satisfies rαi = − sin 2παi cos 2παi a diophantine condition. Herman [He6] stated the following conjecture. Problem 2.2.1 (M. Herman) Show that there exists ε0 > 0 such that in the ball z ≤ ε0 there is a set of positive Lebesgue measure of invariant Lagrangian tori. Note that here the assumption of f being real analytic is essential since in the C ∞ case the conjecture is true for n = 1, open if n = 2 but one knows the existence of counterexamples if n ≥ 3. Using Birkhoff normal form one can prove that the conjecture is true in many special cases (some twist condition, see [BHS]). 2.3

Zk -actions

Let k ≥ 1 and F1 , . . . , Fk be commuting diffeomorphisms in Dr (T). Let α1 , . . . , αk be the rotation numbers of F1 , . . . , Fk . Then the diffeomorphisms f1 , . . . , fk of the circle induced by F1 , . . . , Fk generate a Zk -action on T. If α1 is irrational and F1 = Rα1 , then we must have Fi = Rαi for 1 ≤ i ≤ k, because the centralizer of Rα1 in D0 (T) is the group of translations. Therefore, if α1 is irrational and F1 is conjugated to Rα1 by a diffeomorphism h ∈ Ds (T), the full Zk -action is linearized by h. J. Moser [M2] has shown that, if for some γ, τ and all q > 0 max qαi ≥ γq −τ

1≤i≤k

then there exists a neighborhood V of the identity in D∞ (T) such that if Fi ∈ D∞ (T), ρ(Fi ) = αi and Fi ◦ R−αi ∈ V then the action is linearizable in D∞ (T). The simultaneous approximation condition above is probably optimal in the C ∞ category.

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More generally, one can define, for 1 ≤ s ≤ r ≤ ∞, or r = s = ω k Cr,s = {(α1 , . . . , αk ), any k-uple (F1 , . . . , Fk )

of commuting diffeomorphisms in (Dr (T))k with rotation numbers (α1 , . . . , αk ) is linearizable in Ds (T)}, k,loc if one assumes furthermore that Fi ◦ R−αi , 1 ≤ i ≤ k, and similarly Cr,s r are C -close to the identity. k k,loc Problem 2.3.1 Determine Cr,s , Cr,s . Progress in this direction is contained in the papers [DL], [Kra], [PM], etc. The previous problem generalizes to the study of Zk actions on Rn . Let k > n n ≥ 1 and consider an action Zk → Diffω + (R ). Among them the actions by translations Rαi : x → x+αi , 1 ≤ i ≤ k, where αi ∈ Rn , play a distinguished role. Assume that α1 , . . . , αk generate Rn and consider those actions whose n ω generators f1 , . . . , fk ∈ Diffω + (R ) are C -close to Rα1 , . . . , Rαk and which 0 are C -conjugated to it (thus one can call α1 , . . . , αk the rotation numbers of the action. Problem 2.3.2 For which rotation numbers is this C 0 -conjugacy indeed analytic?

2.4

Diffeomorphisms of compact manifolds

Let M be a C ∞ compact connected manifold. We denote Diff∞ (M ) the group of C ∞ diffeomorphisms of M with the C ∞ topology and Diff∞ + (M ) the group of C ∞ diffeomorphisms C ∞ -isotopic to the identity (i.e. the connected component of Diff∞ (M ) which contains the identity). The general problem that one may address is to study the structure (conjugacy classes, centralizers, etc. ) of Diff∞ (M ). In the special case of the ndimensional torus Tn = Rn /Zn one has a KAM theorem which describes the n local structure of Diff∞ + (T ) near diophantine translations Rα : x → x + α, n n α ∈ T : there exists a neighborhood Uα of Rα in Diff∞ + (T ) such that if ∞ n n f ∈ Uα there exists λ ∈ T and g ∈ Diff+ (T ) such that g(0) = 0 and f = Rλ ◦ g −1 ◦ Rα ◦ g. Moreover this decomposition is locally unique. In [He1] a “converse” of this theorem is asked: Problem 2.4.1 (M. Herman) Let V be a compact C ∞ manifold, f ∈ Diff∞ (V ), U a C ∞ neighborhood of the identity, Of,U = {g ◦ f ◦ g −1 , g ∈ U }. If Of,U is a finite codimension manifold is it true that V = Tn and f is C ∞ conjugate to a Diophantine translation ? In the torus case one proves KAM theorem by means of an implicit function theorem in Fréchet spaces (see [Ha, Bo]). The main point is that a translation Rα being diophantine is equivalent to ask that for all ϕ ∈ C ∞ (Tn ) there exist ψ ∈ C ∞ (Tn ) and λ ∈ R such that the linearized conjugacy equation ψ ◦ Rα − ψ = ϕ + λ


181

holds. Then one can ask the analogue of Problem 2.4.1 for the linearized conjugacy equation Problem 2.4.2 (M. Herman) If for all ϕ ∈ C ∞ (V ) there exists ψ ∈ C ∞ (V ) and λ ∈ R such that ψ ◦ f − ψ = ϕ + λ is it true that V = Tn and f is C ∞ conjugate to a Diophantine translation ?

3 3.1

KAM Theory and Hamiltonian Systems Twist maps

We let T denote the circle R/Z and θ(mod 1) denote the standard parameter of T and x the corresponding parameter of its universal cover R. We will let y ∈ R denote the standard parameter of the second factor of T × R. Let U be an open subset of T × R which intersects each vertical line {θ} × R in an open non empty interval. We consider maps f which are diffeomorphisms from U onto an open subset f (U ) ⊂ T × R which also intersects each vertical line in an open non empty interval. We assume that f is orientation preserving and area preserving. Since we are in two dimensions the area preserving condition is the same as requiring that f be symplectic. Let f˜ denote the lift of f to the universal cover so that f˜(x + 1, y) = f˜(x, y) + (1, 0). We also set (x , y ) = f˜(x, y). An orientation preserving symplectic C 1 diffeomorphism f satisfies a pos itive (resp. negative) monotone twist condition if ∂x ∂y > ε (resp. < −ε) for some fixed ε > 0 and for all (x, y). Geometrically this condition states that the image of a segment x =constant under f˜ forms a graph over the x axis. An integrable twist map has the form f˜(x, y) = (x + r(y), y). From the area preserving property of f it follows that y dx − ydx is a closed 1-form and therefore there exists a generating function (or variational principle) h = h(x, x ) such that y = −∂1 h(x, x ) , y = ∂2 h(x, x ) . The generating function is unique up to the addition of a constant and its invariance under translations (x, x ) → (x + 1, x + 1) is equivalent to the condition that y dx − ydx is exact on T × R. Moreover, from the positive twist condition one has ∂12 h(x, x ) < 0. A rotational invariant curve is a homotopically non-trivial f -invariant curve. By Birkhoff’s theory (see [He3], Chapitre I), such a curve is the graph of a Lipschitz function. For near–to–integrable twist maps KAM theory provides the existence of many rotational invariant curves. Herman [He3, He5] proved that rotational invariant curves persist in twist diffeomorphisms which are C 3 -close to an integrable map. Problem 3.1.1 (J. Mather) Does there exist an example of a C r twist area– preserving map with a rotational invariant curve which is not C 1 (separate question fo each r ∈ [1, ∞] ∪ {ω}).

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Problem 3.1.2 (J. Mather) Given a C ∞ twist diffeomorphism and a rotational invariant curve which is not C ∞ is it possible to destroy it by an arbitrarily small C ∞ perturbation? One knows, following [Ma3] that if the rotation number is not diophantine this is indeed possible even in the case the circle is C ∞ . [Ma3] contains also destruction results for C r twist maps (see p. 212) and [Fo] for analytic maps but in both cases there is a gap between the destruction results and the persistence results given by KAM theory. It is a classical counterexample of Arnold that there exist analytic diffeomorphisms of the circle, with irrational rotation number, whose conjugacy to a rigid rotation is not absolutely continuous. Since every diffeomorphism of the circle can be embedded as rotational invariant curve of an area–preserving monotone twist map of the annulus with the same degree of smoothness, this example, and those constructed by Denjoy [De1, De2] and Herman [He2] in the differentiable case, give examples of “regular” twist maps f having “regular” rotational invariant curves γ such that f |γ is topologically conjugate to a rigid rotation but the conjugacy is not absolutely continuous. In these examples the irrational rotation number has extremely good approximations by rational numbers. The classical results of Denjoy on diffeomorphisms of the circle show that given an invariant curve γ, if the rotation number α of f |γ is irrational then one has the following: • if f |γ ∈ C 2 then f |γ is topologically conjugate to Rα and every orbit is dense in γ; • there exist examples of f |γ ∈ C 2−ε , ε > 0, such that no orbit is dense in γ and the limit set of the orbit of every point of γ is the same Cantor set (Denjoy minimal set). Thus even if f is smooth, limit sets different from γ may appear provided that the invariant curve γ loses smoothness. Problem 3.1.3 (J. Mather) Does there exist a C 3 area–preserving twist map of the annulus with a rotational, not topologically transitive, invariant curve of irrational rotation number? Same problem with r ≥ 3 or even analytic. M. Herman [He3] gave an example of class C 3−ε . Hall and Trupin [HT] gave a C ∞ example without the area–preserving condition. The most important progress towards the understanding of these problems has come through the introduction of Aubry–Mather sets [AL, Ma1, Ma2, Ma4]. These are closed invariant sets given by a parametric representation x = u(θ) , y = −∂1 h(u(θ), u(θ + α)) , where u is monotone (but not necessarily continuous) and u − θ is Z-periodic. They do exist for all rotation numbers α and they are subsets of a closed Lipschitz graph. For rational α one obtains periodic orbits, whereas for irrational


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numbers one has rotational invariant curves if u is continuous (in fact Lipschitz by Birkhoff’s theorem) or an invariant Cantor set if u has countably many discontinuities. In this case the Aubry–Mather set can be viewed as a Cantor set drawn on a Lipschitz graph. Another important property of Aubry–Mather sets is that the “ordering” of an orbit is the same as for the rotation by α of a circle. Mather based his proof on the variational problem 1 h(u(θ), u(θ + α))dθ 0

minimizing this functional in the class of weakly monotone functions, thus they are also called action minimising sets. Problem 3.1.4 (M. Herman) For a C r twist area-preserving diffeomorphism does the union of the action-minimising sets which are not closed curves and do not contain periodic orbits have Hausdorff dimension 0? Here r ≥ 3; otherwise Herman himself has a counterexample. 3.2

Euler–Lagrange flows

It is proved in [M1] that any monotone twist map can be obtained as the time1 map of the Hamiltonian flow associated to a time–dependent, Z-periodic in time Hamiltonian H : T ∗ (T × R) × R → R satisfying Legendre condition Hyy (θ, y, t) > 0. This assures that f can also be interpolated by the time1 map of the Euler–Lagrange flow associated to the Lagrangian function L : T (T × R) × R → R obtained from H by Legendre transform. More generally, let M be a a closed Riemannian manifold M and consider Lagrangians of the form kinetic energy + time periodic potential V ∈ C r (M × T) (see [Ma5] for a more general setting). Assume that M has dimension at least 2. Problem 3.2.1 (J. Mather) Is there a residual set (in the sense of Baire category) in C r (M × T) such that there exists a corresponding trajectory of the Euler–Lagrange flow with kinetic energy growing to ∞ as t → ∞? Of course these systems are very far from integrable ones. De La Llave has some results concerning this problem. 3.3

n-body problem

Let n ≥ 2. Consider n + 1 point masses m0 , . . . , mn moving in an inertial reference frame R3 with the only forces acting on them being their mutual gravitational attraction. If the i-th particle has position vector qi then the Newtonian equations of motion are mi qï = −

∂V , i = 1, . . . , n ∂qi

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Gmi mj where V (q0 , . . . , qn ) = − 0≤i<j≤n |qi −q and G is the universal gravitaj| tional constant. The size of the system is measured by the moment of inn ertia I = 12 i=0 mi |qi |2 . The total energy of the system E = T + V = n n |qi |2 angular momentum A = i=0 mi qi ∧ q˙i i=0 mi 2 +V (q0 , . . . qn ), the total n and the total momentum P = i=0 mi q˙i are integrals of the motion. We will n assume that the center of mass C = i=0 mi qi is fixed at the origin, thus P = 0. J. Mather recalled a problem proposed by G.D. Birkhoff [Bi]: Problem 3.3.1 (G.D. Birkhoff ) Let n = 2 (three body problem) and assume that the total energy is negative. One can even assume that the three particles move on a fixed plane. Is it true that the moment of inertia of the system can grow to ∞ as t → ∞ for a dense open set of initial conditions? One can ask the same question for n > 2. The restriction to negative energy is necessary since from the identity of Jacobi–Lagrange I¨ = E + T follows that if the energy E ≥ 0 all orbits which are defined for all times are wandering. The only thing which is known is that even for negative energy wandering sets do exist (as Birkhoff and Chazy showed long ago, see [Al]). The fact that the bounded orbits form a positive Lebesgue-measure set for the planar three-body problem is a consequence of KAM theory (see [Ar]). According to M. Herman [He6] “what seems not an unreasonable question to ask (and possibly prove in a finite time with a lot of technical details) is” Problem 3.3.2 (M. Herman) If one of the masses m0 = 1 and all the other masses mj 0 H0s (T1 ) (here H0s (T1 ) is the Sobolev space formed by zero-meanvalue functions on T1 ) contains a smooth invariant 2n-dimensional manifold T 2n such that


187

(a) the restriction of (KdV) to T 2n defines a Liouville–Arnold integrable Hamiltonian system; (b) T 2n ⊂ T 2m if m > n; (c) ∪T 2n is dense in each H0s (T1 ). Moreover the invariant manifolds T 2n are filled with time-quasiperiodic solutions (the so-called n-gap solutions, see the lectures of S. Kuksin in this volume). Also the Sine–Gordon (SG) equation under Dirichlet boundary conditions utt = uxx − sin u,

u(t, 0) = u(t, π) = 0,

(SG)

is integrable (i.e. it has n-gap solutions) but the manifolds T 2n have algebraic singularities and their union is proved to be dense only near the origin. We refer to [B] and [Ku] for an introduction to the recent progress on the theory of nonlinear Hamiltonian PDEs. Problem 4.3.1 (S. Kuksin) Find a Lyapounov Theorem (not of KAM type) for Hamiltonian PDEs. I.e. prove that (under reasonable assumptions) most of time-periodic solutions from any non-degenerate one-parameter family persist under small Hamiltonian perturbations of the equation. For further information and some progress in this direction see [Ba2]. Problem 4.3.2 (S. Kuksin) Find a non-perturbative way to obtain timeperiodic solutions of an Hamiltonian PDE (the solutions should not be travelling waves). The main object of the Lectures of Kuksin in this volume is the proof of a KAM-type theorem wich assures the persistence under small Hamiltonian perturbations of most finite-gap solutions of Lax-integrable Hamiltonian PDEs like (KdV) or (SG). Nothing is known on the infinite-gap solutions (almost periodic solutions, see [AP] for an introduction). Following [MT] (see also [BKM]), one can ask: Problem 4.3.3 (S. Kuksin) Any Sobolev space H0m (T1 ), m ≥ 1, is filled by almost periodic solutions of (KdV). Do most of them persist under small Hamiltonian perturbations of the equation ? Presumably they do not. From the KAM theorem described in Kuksin’s lectures it also follows [BK] that most small amplitude finite-gap solutions of the ϕ4 equation utt = uxx − mu + γu3 , m, γ > 0 ,

(ϕ4 )

persist under small Hamiltonian perturbations. The (ϕ4 ) equation can be indeed obtained from the (SG) equation developing sin u into Taylor series at the third order and truncating. If one sets m = 0 into (ϕ4 ) the existence of small-amplitude time-quasiperiodic solutions is not known. Problem 4.3.4 (S. Kuksin) Construct small amplitude time quasiperiodic solutions for the massless ϕ4 equation utt = uxx − u3 , x ∈ T1 .

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Let us consider again the (NS) equation. As we have seen the time-flow preserves the Hamiltonian and the L2 -norm of the solutions. One can study the following properties of the long-time behaviour of the solutions: (a) (a’) (b) (b’)

limt→∞ u(t)m = ∞ (here m is sufficiently large); lim supt→∞ u(t)m = ∞; u(t) → 0 weakly in H s as t → ∞; u(tn ) → 0 weakly in H s for a suitable sequence tn → ∞.

Problem 4.3.5 (S. Kuksin) Which properties among a)–b’) hold for the solutions of (NS) with a typical initial condition u0 , u(0, x) = u0 (x) ∈ H m ? Note the following facts: • (a)–(b’) are all wrong for any u0 if p = 1, n = 1; • (a)–(b’) are wrong for some u0 (for any p, n) since (NS) has time-periodic solutions; • (b) ⇒ (a) and (b’) ⇒ (a’) due to the conservation of the L2 -norm. Problem 4.3.6 (S. Kuksin) Does (NS) have an invariant measure in H s such that its support (in H s ) is a set with non-empty interior (here s is sufficiently large) ? Not that a positive answer to Problem 4.9 implies that a) is wrong almost everywhere with respect to the measure. Problem 4.3.7 (S. Kuksin) In the spirit of Nekhoroshev’s theorem one can ask the following: does an arbitrary solution of a ε-perturbed KdV equation remain εa -close to some finite gap torus during a time interval 0 ≤ t ≤ Tε with Tε ≥ CM ε−M for all M > 0? Probably a crucial step is to prove this statement for T = ε−2 . A first result in this direction has been obtained in [Ba1].

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Dynamical systems and small divisors: Lectures CIME school

Six lectures on dynamical systems

Lectures on chaotic dynamical systems

Lectures on chaotic dynamical systems

Optimization and dynamical systems

Optimization and Dynamical Systems

Dynamical systems and processes

Dynamical systems and chaos

Dynamical Systems and Methods

Geometric topology: recent developments. Lectures CIME